A complete classification of cubic function fields over any finite field
aa r X i v : . [ m a t h . N T ] A p r A COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITEFIELD
SOPHIE MARQUES AND KENNETH WARD
Abstract.
We classify all cubic function fields over any finite field, particularly developing a complete Galoistheory which includes those cases when the constant field is missing certain roots of unity. In doing so, we findcriteria which allow one to easily read ramification and splitting data from the generating equation, in analogyto the known theory for Artin-Schreier and Kummer extensions. We also describe explicit irreducibility criteria,integral bases, and Galois actions in terms of canonical generating equations.MSC Code (primary): 11T22MSC Codes (secondary): 11T55, 11R32, 14G15, 11R58Keywords: Cyclotomy, cubic, function field, finite field, Galois
Contents
Introduction 1Acknowledgements 21. Minimal polynomials for generators of cubic extensions 22. Galois closure of cubic extensions 63. Cubic extensions in characteristic
84. Purely cubic extensions and Kummer extensions 105. Extensions with generating equation y ´ y ´ a “ X ´ X ´ a “ Introduction
Let p ą be a prime integer, F q a finite field with q “ p n elements, and K a function field withfield of constants F q . Let L { K be a Galois cubic extension. In general, if the field characteristic isequal to 3, then Artin-Schreier theory is used to describe the extension L { K , which is given by anequation y ´ y “ a , for a P K , whereas if the characteristic is not equal to 3 and the constant fieldcontains a primitive third root of unity, then Galois theory is well understood via Kummer theory,which gives a generating equation y “ a , for a P K (see for example [7, 8]). The situation is moredelicate when the constant field does not contain a primitive third root of unity. Our goal in this paperis to investigate all Galois cubic extensions of function fields K over finite fields in any characteristic, and in doing so, we are able to give a canonical Galois theory for cubic function fields over finite fieldswhen Artin-Schreier and Kummer theory cannot be used.We refer to extensions with generating equation X “ a as purely cubic . If the characteristic isdifferent from , we give a simple criterion which determines whether or not a given extension offunction fields is purely cubic. If such an extension L { K is not purely cubic, then we show that it hasa generation of the form L “ K p y q , with y such that y ´ y ´ a “ p a P K q . We will study this form in detail, as it is central to Galois theory when Artin-Schreier and Kummertheory are not useful. We show that it is possible, and in fact practical, to read the ramificationand splitting directly from this form. In particular, this may be done in terms of the factorisationof a . This gives a version of standard form , which is well-known for Kummer and Artin-Schreierextensions (and we will henceforth refer to our form as standard due to this analogy). We are alsoable to completely describe the following using our standard form in conjunction with Artin-Schreierand Kummer theory:— Irreducibility criteria for degree 3 polynomials;— Galois cubic extensions of function fields in any characteristic;— Galois actions and their connections to splitting; and— Algorithms for providing integral bases.We emphasise that the advantage of this approach is the ability to obtain all of the above informationconcisely using our standard form. When joined with classical Artin-Schreier and Kummer theory, theresults which follow therefore provide a complete study of Galois structure for cubic function fieldsover finite fields.For ease of reading, we have divided this paper in the following way. Section 1 is devoted toreducing general minimal polynomials of cubic extensions L { K to standard forms. In this way, werecover Artin-Schreier and Kummer theory. For completeness, Section 2 examines the structure ofthe Galois closure of L { K when the extension is not necessarily Galois. Section 3 reviews the Artin-Schreier and its ramification theory, and Section 4 does just the same for Kummer extensions. Section5 addresses the matter of cubic extensions of function fields over finite fields when Artin-Schreier andKummer theory cannot be used, which we note is the crux and raison d’ˆetre of this paper. Owing tothe length of the arguments, we relegate a portion of the full proofs of Section 5 to the Appendix. Acknowledgements
The authors thank Ben Blum Smith for valuable comments and discussions on Theorem 5.1.Kenneth Ward thanks the CAS Mellon Fund at American University for its generous support.1.
Minimal polynomials for generators of cubic extensions
In this section, we prove that any cubic extension has a primitive element whose minimal polynomialis either of the form(1) T p X q “ X ´ a , for some a P K or(2) T p X q “ X ` X ` b for some b P K , and that this is equivalent to the existence of a generatorwhose minimal polynomial is of the form(a) T p X q “ X ´ X ´ a , for some a P K , in characteristic different from .(b) T p X q “ X ` aX ` a , for some a P K , in characteristic .We note that the first case yields a purely inseparable extension in characteristic . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 3
Definition 1.1.
A cubic extension L { K is called purely cubic if there exists a primitive element y for L { K such that the minimal polynomial of y over K is of the form y “ a , with a P K . If the characteristic is different from , we will give a simple criterion (Corollary 1.4) which de-termines whether or not a given extension of function fields is purely cubic. The following Lemmaproves that one can find a generator with no square term. This is well known and originally due toTartaglia, as in his contribution to the Ars Magna , “On the cube and first power equal to the number”[10, Chapter XV, p. 114]. (For a more modern reference, see also [6, p. 112].)
Lemma 1.2.
Suppose that p ‰ , and let L { K be a separable extension of degree . Then there existsan explicit primitive element y with characteristic polynomial of the form T p X q “ X ` aX ` b , with a , b P K . The discriminant of the polynomial T p X q “ X ` aX ` b of Lemma 1.2 is equal to ´ a ´ b . Wewill use this form of the discriminant later. In the following Theorem, we show that one may finda generator of the form announced at the outset of this section. This will be crucial in the sequel;one major goal of this analysis is to use the generator of a cubic extension to study ramification,reducibility, and integral basis construction. Theorem 1.3.
Let p ‰ . Let L { K be a separable extension of degree with generating equation T p y q “ y ` ey ` f y ` g “ , where e , f , g P K .(a) Suppose that eg ‰ f . Then there exists a primitive element z of L { K with characteristic polyno-mial of the form T p X q “ X ` X ` b , with b P K .(b) Suppose that eg “ f . Then there exists a primitive element z of L { K with characteristic polyno-mial of the form T p X q “ X ´ b , with b P K .Furthermore, in each case, this primitive element is explicitly determined.Proof. (a) Let y a generating element of L { K then there exist e , f , g P K , g ‰ such that the minimalpolynomial of y is of the form: S p X q “ X ` eX ` f X ` g This may be seen by performing the rational transformation y Ñ y “ y f g y ` , as g ‰ by irreducibility of S p X q . It follows that y “ y ´ f y g , which yields ¨˝ y ´ f y g ˛‚ ` e ¨˝ y ´ f y g ˛‚ ` f ¨˝ y ´ f y g ˛‚ ` g “ . Multiplication by p ´ f y g q then yields “ y ` ey ˆ ´ f y g ˙ ` f y ˆ ´ f y g ˙ ` g ˆ ´ f y g ˙ “ ˜ ´ e f g ` f g ¸ y ` ˜ e ´ f g ¸ y ` g . (1) SOPHIE MARQUES AND KENNETH WARD
Let γ “ ´ e f g ` f g . If f “ , then γ “ ‰ , and otherwise, γ “ ` e ˆ ´ f g ˙ ` f ˆ ´ f g ˙ ` g ˆ ´ f g ˙ “ ˆ ´ f g ˙ ˜ˆ ´ gf ˙ ` e ˆ ´ gf ˙ ` f ˆ ´ gf ˙ ` g ¸ “ ˆ ´ f g ˙ S ˆ ´ gf ˙ . As the polynomial S p X q “ X ` eX ` f X ` g is irreducible over K , it follows that ´ gf cannot bea root of S , whence γ “ ˆ ´ f g ˙ S ˆ ´ gf ˙ ‰ . Therefore, in any case, γ ‰ . We let α “ e ´ f g . Dividing (1) by γ , we obtain(2) y ` αγ y ` g γ “ . By assumption, eg ‰ f , so that α ‰ . We let z “ γα y . Hence, with b “ γ g α “ ´ ´ e f g ` f g ¯ g ´ e ´ f g ¯ “ p g ´ e f g ` f q p ge ´ f q and this choice of z , we obtain z ` z ` b “ , as desired.(b) This follows as in the previous case by (2) and b “ ´ g γ . (cid:3) We note that the discriminant of the polynomial T p X q of Lemma 1.3(a) is equal to ´ b ´ b . Corollary 1.4.
Let p ‰ . Let L { K be a separable extension of degree with generating equation T p y q “ y ` ey ` f y ` g “ , where e , f , g P K .(a) Suppose that eg ‰ f . Then there exists a primitive element z for L { K with characteristic poly-nomial of the form T p X q “ X ´ X ´ b , with b P K .(b) Suppose that eg “ f . Then there exists a primitive element z for L { K with characteristic poly-nomial of the form T p X q “ X ´ b , with b P K .In each case, this primitive element is explicitly determined.Proof. (a) By Lemma 1.3, if eg ‰ f , then there exists an explicitly determined primitive element z for L { K with characteristic polynomial of the form P p X q “ X ` X ` a , with a P K . We thenperform the linear change of variable z “ z ´ and “ ˆ z ´ ˙ ` ˆ z ´ ˙ ` a “ z ´ z ` z ´ ` z ´ z ` ` a COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 5 “ z ´ z ` ` a . Letting z “ z , we therefore obtain “ «ˆ z ˙ ´ z ` ` a ff “ z ´ z ` ` a . Letting b “ ´ ´ a “ ´ ´ p g ´ e f g ` f q p ge ´ f q , it follows that z is the desired primitive element.(b) This case follows immediately from Lemma 1.3(b). (cid:3) We note that the discriminant of the polynomial P p X q of Corollary 1(a) is equal to ´ p´ ` b q .If the characteristic is equal to , we find two forms for the minimal polynomial of a generator of acubic extension: Theorem 1.5.
Suppose the characteristic of k is . Let L { K be a extension of degree . Then either:(1) L { K is separable, there is a primitive element z such that its minimal polynomial is equal to T p X q “ X ` X ` b , and this primitive element is explicitly determined; or(2) L { K is purely inseparable and there is a primitive element z such that its minimal polynomialis equal to T p X q “ X ` b .Proof. Let y a generating element of L { K then there exist e , f , g P K , g ‰ such that the minimalpolynomial of y is of the form: S p X q “ X ` eX ` f X ` g Clearly, L { K is separable if, and only if, not both e and f are zero.Suppose then that L { K is separable. If e “ , then as T p y q is irreducible, g ‰ . Let y “ { y . Then y ` fg y ` g “ . If L { K separable then f ‰ , and taking z “ gf y , we obtain z ` z ` g f “ . If, on the other hand, e ‰ , then let y “ y ´ fe . As the field characteristic is equal to 3, then “ ˆ y ` fe ˙ ` e ˆ y ` fe ˙ ` f ˆ y ` fe ˙ ` g “ y ` f e ` ey ` f y ` f e ` f y ` f e ` g “ y ` ey ` f e ` g ` f e . With z “ e y , we therefore obtain “ z ` z ` f e ` ge ` f e . SOPHIE MARQUES AND KENNETH WARD (cid:3)
Corollary 1.6.
Let p “ . Let L { K be a separable extension of degree . Then there is a primitiveelement z such that its minimal polynomial is of the form T p X q “ X ` bX ` b . Furthermore, thisprimitive element is explicitly determined.Proof. From the previous Theorem, there is a primitive element y such that y ` y ` b “ . Let y “ { y ; then by ` y ` “ . Finally, letting z “ by , we obtain z ` bz ` b “ . (cid:3) Galois closure of cubic extensions
We begin this Section with a well-known result on the Galois group of a cubic extension (see forinstance [1, Theorems 1.1, 2.1, 2.6]):
Lemma 2.1.
Let p ‰ . Suppose that the cubic extension L { K is separable. Then L { K is Galois if,and only if, the discriminant is a square in K . Furthermore, in this case, Gal p L { K q “ A . We denote the minimal polynomial of a primitive element y of L { K as T p X q “ X ` eX ` f X ` g , e , f , g P K . As T p X q is irreducible, it is of course necessary that g ‰ . As T p X q has degree in X , it follows that T p X q is irreducible over K if, and only if, it possesses no root in K . For ease of notation, we denoteby L G K { K the Galois closure of L { K . We have Gal p L G K | K q E S , as the Galois group G K permutes theroots of the minimal polynomial of a primitive element of L { K . The following two results are generaland hold for any characteristic; these give the explicit construction of the Galois closure (Ibid.). Lemma 2.2.
Let L { K be a separable extension of degree with generating equation y ` ey ` f y ` g “ , where e , f , g P K with roots α, β, γ in the splitting field L s of L . The two elements of L s α β ` β γ ` γ α and β α ` α γ ` γ β are roots of the polynomial R p X q “ X ` p e f ´ g q X ` p e g ` f ` g ´ e f g q , which has the same discriminant as T p X q “ X ` eX ` f X ` g . The quadratic function R p X q is called the quadratic resolvent of T p X q . Theorem 2.3.
Suppose T p X q P K r X s is a separable irreducible cubic polynomial which defines a cubicfield extension L { K . Then L G K { K has automorphism group over K equal to:(a) A if its quadratic resolvent R p X q is reducible over K ; and(b) S if its quadratic resolvent R p X q is irreducible over K . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 7
Furthermore, in case (b), the Galois closure L G K of L { K is equal to K p α, δ q , where α is any root of T p X q ,and δ is either root of the quadratic resolvent of T p X q . If p ‰ , then L G K is also equal to K p α, ? D q ,where D is the discriminant of T p X q . We now suppose that L { K is a separable but non-Galois extension. ˝ Suppose moreover that the field characteristic is not or ; then either ˝ L { K is purely cubic and the Galois closure is equal to K p y , ξ q , where y generates a cubicextension whose generating equation is X ´ a and ξ is a primitive third root of unity. L G K y “ a ❊❊❊❊❊❊❊❊❊ ξ ` ξ ` “ ②②②②②②②②② K p y q y “ a ❊❊❊❊❊❊❊❊❊ K p ξ q ξ ` ξ ` “ ②②②②②②②②② K Note that K p ξ q{ K is a constant extension, the constant field of L G K is F q p ξ q , and L G K { K p ξ q isa Kummer extension. ˝ L { K has a primitive element y with minimal polynomial of the form X ´ X ´ a , for some a P K and the Galois closure is K p y , γ q with γ “ ´ p a ´ q . L G K y ´ y “ a ❊❊❊❊❊❊❊❊❊ γ “´ p a ´ q ②②②②②②②②② K p y q y ´ y “ a ❊❊❊❊❊❊❊❊❊ K p γ q γ “´ p a ´ q ①①①①①①①①① K Note that γ “ ´ p a ´ q is a Kummer extension, and thus the ramified places are those p such that p v p p´ p a ´ qq , q “ . Thus, any place p of K which ramifies in K p γ q must satisfy v p p a q “ . Moreover, we will prove later that the ramified places of y ´ y “ a are the places p such that v p p a q ă and ∤ v p p a q (see Theorem 5.10). In particular, there are no places fullyramified in the Galois closure L G K . ˝ If the field characteristic is equal to , then L { K has a primitive element y whose minimalpolynomial is X ´ aX ´ a for some a P K . Then the Galois closure L G K { K of L { K is L p z , γ q where γ “ a , and z “ γ y generates a cubic Artin-Schreier equation with generating equation z ´ z “ γ . (See Theorem 3.1) L G K z ´ z “ γ ❊❊❊❊❊❊❊❊❊ γ “ a ②②②②②②②②② K p y q y ´ ay “ a ❊❊❊❊❊❊❊❊❊ K p γ q γ “ a ①①①①①①①①① K SOPHIE MARQUES AND KENNETH WARD
We note that K p γ q{ K is Kummer extension, and that L G K { K p γ q is an Artin-Schreier extension. ˝ If the field characteristic is equal to , then either ˝ L { K is purely cubic and the Galois closure is L p y , ξ q , where y generates a cubic extensionwhose generating equation is X ´ a and ξ is a primitive third root of unity. Indeed, thequadratic resolvent is X ` aX ` a and ξ a is a root of this polynomial. L G K y “ a ❊❊❊❊❊❊❊❊❊ γ ` a γ ` a “ ②②②②②②②②② K p y q y “ a ❊❊❊❊❊❊❊❊❊ K p ξ q ξ ´ ξ “ ②②②②②②②②② K Note again that K p ξ q{ K is a constant extension, whence the constant field of L G K is equal to F q p ξ q . ˝ L { K has a primitive element y with minimal polynomial of the form P p X q “ X ´ X ´ a forsome a P K and the Galois closure is equal to L p y , γ q , where γ is an Artin-Schreier generatorof degree whose minimal polynomial is X ´ X “ ` a a . Indeed, the quadratic resolvent of P p X q is equal to X ` aX ` p ` a q . L G K y ´ y “ a ❊❊❊❊❊❊❊❊❊ z ` az `p ` a q“ ②②②②②②②②② K p y q y ´ y “ a ❊❊❊❊❊❊❊❊❊ K p γ q γ ´ γ “ ` a a ①①①①①①①①① K We note that K p γ q{ K is an Artin-Schreier extension.3. Cubic extensions in characteristic As we have seen in Section , separable cubic extensions of function fields in characteristic haveprimitive elements with minimal equation of the form T p X q “ X ` bX ` b . If the extension L { K isGalois, this extension is Artin-Schreier as is well-known. We reprove this basic result in Galois theoryusing our previous standard form. Theorem 3.1.
Let p “ , and let L { K be a Galois extension of degree . Then there is a primitiveelement z such that its minimal polynomial is of the form R p z q “ z ´ z ´ a . Furthermore, this primitiveelement is explicitly determined.Proof. By the previous result, we know that there is a primitive element y such that its minimalpolynomial is of the form S p X q “ X ` bX ` b . The discriminant of such a polynomial is equal to ´ b . As L { K is Galois, the discriminant is a square,thus ´ b is a square, say ´ b “ a . With z “ y { a , it follows that z ´ z “ a . (cid:3) COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 9
Although the Artin-Schreier theory is well-known, we will draw certain important parallels to itsfundamental results in other types of extensions, particularly those of the form y ´ y “ a whichoccur when q ” ´ . For the proof of the following classical result, we refer the reader to [8,Proposition 5.8.6]. Lemma 3.2.
Let char p F q q “ , and let L i “ K p z i q{ K p i “ , q be two cyclic extensions of degree ofthe form z i ´ z i “ a i P K , i “ , . Then the following statements are equivalent:(1) L “ L .(2) z “ jz ` b for ď j ď and b P K .(3) a “ ja ` p b ´ b q for ď j ď and b P K . The following Theorem gives ramification data, genus and Galois action for Artin-Schreier extensions(see [7, Proposition 3.7.8]).
Theorem 3.3.
Let K { F q be an algebraic function field of characteristic . Suppose that a P K is anelement which satisfies the condition a ‰ w ´ w f or all w P K . Let L “ K p y q with y ´ y “ a . Such an extension L { K is called an Artin-Schreier extension of K . Wethen have:(1) L { K is a cyclic Galois extension of degree . The automorphisms of L { K are given by σ l p y q “ y ` l ( l “ , , ).(2) A place p of K is ramified in L if, and only if, there is an element z P K satisfying v p p a ´ p z ´ z qq “ ´ m p ă and m p ı . For a place p of K ramified in L , let P denote the unique place of L lying over p . Then thedifferential exponent α p P | p q is given by α p P | p q “ p m p ` q . (3) If at least one place Q satisfies v Q p a q ą , then F q is algebraically closed in L and g L “ g K ´ ` ÿ p ramified in L p m p ` q deg p p q , where g L (resp. g K ) is the genus of L { F q (resp. K { F q ) and deg p p q is the degree of the place p of K . Remark 3.4.
Suppose p “ . Let L { K be a cubic extension. By Corollary 1.6, we know that there isa primitive element z for L with minimal polynomial T p X q “ X ´ bX ´ b , with b P K . By the resultsof Section 2, we know that the Galois closure L G K { K can be expressed as a tower L G K “ K p y , β q{ K p β q{ K with y that has minimal polynomial S p X q “ X ´ X ´ β where β “ b . As r K p y , β q : K p β qs “ iscoprime to r K p β q : K s “ , the ramified places of L { K are the places of K that ramify in K p y , β q{ K p β q .These are precisely those places p such that there is place P in K p β q above p and an element a P K p β q such that v p p β ´ p a ´ a qq “ ´ m ă and gcd p m , q “ . If p “ , for a Galois cubic extension L { F q p x q , it is known (see [8, Example 5.8.8]) via a processoriginally due to Hasse that a generating equation of the form P p X q “ X ´ X ´ a may be transformedinto another generating equation X ´ X ´ c where the ramified places of L { F q p x q are given by theplaces p of F q p x q for which v p p c q ă , and for each such place, p v p p c q , q “ . In this case, the equation X ´ X ´ c is said to be in standard form .We recall when an Artin-Schreier extension is constant. Theorem 3.5.
Suppose that p “ , L { F q p x q is cubic and Galois, it is thus an Artin-Schreier extensionwith a primitive element y of L { F q p x q and generating equation X ´ X ´ c in standard form. Then c P F q if, and only if, L { F q p x q is a constant extension (whence L “ F q p x q ).Proof. If c P F q , then by definition of the generating equation y ´ y ´ c “ , L { F q p x q is obtainedby adjoining constants. For the converse, suppose that c R F q . Then there would exist a place p of F q p x q such that v p p c q ă . As the generating equation is in standard form, it follows that p is (fully)ramified in L . As constant extensions are unramified (see for example [8, Theorem 6.1.3]), it followsthat L { F q p x q cannot be a constant extension. (cid:3) We conclude this section by giving an integral basis for Artin-Schreier extensions [4, Theorem 9].
Theorem 3.6.
Let L be an Artin-Schreier extension of F q p x q with generating equation y ´ y “ a ,where the factorisation of a in F q p x q is given by a “ Q ś li “ P λ i i , where P i P F q r x s and p λ i , q “ for each i “ , . . . , l . (This is known to exist; see [8, Example 5.8.8] .)Then t , S y , S y u is an integral basis of L over F q p x q , where for each j “ , , S j “ l ź i “ P r i , j i , with r i , j “ ` Y j λ i ] , where t x u denotes the integral part of x . Purely cubic extensions and Kummer extensions
In this section, we obtain a criterion for the coefficients of the minimal polynomial of a cubicextension L { K to determine whether or not the extension is purely cubic. Theorem 4.1.
Suppose p ‰ . Let L { K be a cubic extension with constant field F q and a primitiveelement z whose minimal polynomial is T p X q “ X ` eX ` f X ` g where e , f , g P K . If eg ‰ f , L { K admits an explicitly determined primitive element y such that y ´ y “ a where a “ ´ ´ p g ´ e f g ` f q p ge ´ f q . Then, L { K is purely cubic if, and only if, ˝ either eg “ f or a ´ is a square in K , if char p F q q ‰ . (The generator is explicitly deter-mined.) ˝ either eg “ f or the polynomial X ´ X “ a has a root in K , if char p F q q “ .More precisely, then for any extension L { K admitting a primitive element y such that y ´ y “ a with a P K ,(1) If p ‰ and a ´ is a square, say, a ´ “ δ , then u “ y ` ky ´ is such that u “ δ k ,where k “ ´ a ˘ δ . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 11 (2) If p “ and K “ F q p x q , the polynomial X ´ X “ a has a root in K if and only if a “ Z W p Z ` W q for some W , Z P F q r x s with p W , Z q “ and u “ y ` ky is such that u “ W p Z ` W q P , where k “ W p Z ` W q Z . Proof.
From Theorem 1.4, we know that either eg “ f and L { K is purely cubic, or that L { K admitsa primitive element y such that y ´ y “ a with a “ ´ ´ p g ´ e f g ` f q p ge ´ f q P K . The proof that follows does not use this form of a , and is therefore valid for any extension L { K witha primitive element y such that y ´ y “ a .We note that any primitive element is of the form u “ jy ` ky ` l for j , k P K not both equal to and l P K . Thus, we wish to determine which extensions admit a primitive element u such that u “ b ,for some b P K . We have u “p jy ` ky ` l q “ j y ` k j y ` j p l j ` k q y ` k p k ` l j q y ` l p l j ` k q y ` l ky ` l “ k j y ` p j ` jl ` k l ` l j ` k j ` j ak q y ` p j a ` l k ` lk j ` k ` j al ` jak q y ` j a ` ak ` kal j ` l “p j ` jl ` k l ` l j ` k j ` j ak q y ` p k j ` j a ` l k ` lk j ` k ` j al ` jak q y ` j ak ` ak ` kal j ` l ` j a . As , y , y form a basis of L { K , we then have that $&% j ` jl ` k l ` l j ` k j ` j ak “ p q k j ` j a ` l k ` lk j ` k ` j al ` jak “ p q j ak ` ak ` kal j ` l ` j a “ b p q If j were equal to , then from p q , either k “ or l “ , whence from p q , k “ , and thus that u would not be a primitive element for L . Therefore, j ‰ . Without loss of generality, we may supposethat j “ . Thus, we obtain $&% ` l ` k l ` l ` k ` ak “ p q k ` a ` l k ` lk ` k ` al ` ak “ p q ak ` ak ` kal ` l ` a “ b p q Evaluating k ¨ p q ´ p q yields p k ´ k ´ a qp l ` q “ . As k ´ k ´ a ‰ by irreducibility of the polynomial T p X q “ X ´ X ´ a over K , it follows that l “ ´ . By substitution in p q , p q , p q respectively, we obtain $&% ` ka ` k “ p q k p ` ka ` k q “ p q ak ´ ak ` p´ ` a q “ b p q Evaluating a ¨ p q ´ p q , we obtain a k ` ak ´ p´ ` a q ` b “ p q . Via p q ´ a ¨ p q , we find ´ a p a ´ q k ´ p a ´ q ` b “ . Thus, k “ ´ p a ´ q ` ba p a ´ q . By substitution of k “ ´ p a ´ q ` ba p a ´ q in p q , we have(3) p a ´ q ´ b p a ´ q ´ b “ . This is a quadratic in b with discriminant equal to ∆ “ a p a ´ q . (1) If p ‰ , then the quadratic (3) has a solution, if and only if, a ´ is a square, say a ´ “ δ .Then, b “ ´ p a ´ q ˘ a p a ´ q δ , and thus k “ ´ p a ´ q ` ba p a ´ q “ ´ a ˘ δ Note that b “ p a ´ q δ k . (2) If p “ this quadratic, then (3) has a solution, if and only if, a ´ a X ´ X “ has a root b in K , which is equivalent to that X ´ X ´ a has a root c in K with c “ a b . Now,we write c “ CD and a “ PQ with P , Q , C , D P F q r x s and p P , Q q “ and p C , D q “ , then we have ˆ CD ˙ ` CD “ C ` CDD “ C p C ` D q D “ Q P As p C , D q “ , we can assume without loss of generality that C p C ` D q “ Q and D “ P .As p “ , this implies that D “ P and C p C ` P q “ Q . The polynomials C and C ` P arecoprime as p C , D q “ ; as D “ P and F q r x s is a prime factorisation domain, we have that C and C ` P are therefore square, and thus that there exist W and V P F q with p W , V q “ suchthat C “ W and C ` P “ V . As a consequence, P “ W ` V and WV “ Q . Finally, putting Z “ W ` V , we get P “ Z , Q “ W p Z ` W q , C “ W and D “ P , with p Z , W q “ . Hence, c “ W Z , b “ ca “ W Q Z P “ W p Z ` W q P and k “ ba “ W p Z ` W q P . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 13
By construction, if a is of this form, then the polynomial X ´ X ´ a has a root c “ W Z . TheTheorem follows. (cid:3) The following Corollary is also well known. We choose to use the previous Theorem to reprove itdirectly.
Corollary 4.2.
Suppose that K contains a primitive third root of unity. (This is equivalent to q ” .) Then a geometric cubic extension L { K is Galois, if and only if, L { K purely cubic extension.In this case, L { K is called a Kummer extension .Proof.
Suppose that K contains a primitive third root of unity ξ . If L { K is purely cubic, then we mayfind a primitive element y such that its minimal polynomial is y “ a , for some a P K . Clearly, y , ξ y ,and ξ y are the roots of the minimal polynomial of y , and they are all contained in L . Thus, L { K isGalois and Gal p L { K q “ Z { Z .Suppose now that L { K is Galois, and let y be a primitive element of L { K with minimal equation y ` ey ` f y ` g “ . By Theorem 1.4, if eg “ f , then L { K is purely cubic. Suppose then that eg ‰ f . Thus, there exists a primitive element z with minimal polynomial T p X q “ X ´ X ´ a . Thediscriminant of this polynomial is equal to ∆ “ ´ p´ ` a q . ˝ If p ‰ , then as the extension L { K is Galois, the discriminant ∆ is a square in K . As K containsa primitive third root of unity, ´ is a square in K , whence p a ´ q must also be a square in K ,and by the previous Theorem, it follows that L { K is purely cubic. ˝ If p “ , then the quadratic resolvent of T p X q is equal to X ` aX ` p ` a q . As L { K is Galois,this polynomial has a root in K , say, δ . From the previous Theorem, L { K is purely cubic if, andonly if, X ´ X ´ a has a root in K , which is equivalent to S p X q “ X ` aX ` having a root in K . Note that there is a root of S p X q of the form δ ` u , as then S p δ ` u q “ p δ ` u q ` a p δ ` u q ` “ p u ` δ q ` a p δ ` u q ` “ u ` au ` a . The element u “ ξ a is a solution to this equation, where ξ is a root of unity, whence L { K ispurely cubic. (cid:3) Corollary 4.3.
Let p ‰ . A purely cubic extension L { K is Galois if, and only if, K contains aprimitive third root of unity.Proof. Suppose that L { K is purely cubic, so that y “ α for some α P K .(1) Case 1: p ‰ . By Lemma 2.1, L { K is Galois, if and only if, d L { K “ ´ α equal to a square in K . This is equivalent to ´ being a square in K , which in turn is equivalent to K containing aprimitive third root of unity.(2) Case 2: p “ . By Lemma 2.3, the extension L { K is Galois if, and only if, the resolventpolynomial R p X q “ X ` α X ` α is reducible, which is true if, and only if, X ` X ` isreducible. That is, K contains a primitive third root of unity.Thus, in either case, the result follows. (cid:3) We will give the equivalent form of the next result later for extensions with generating equation y ´ y ´ a “ . For this reason, we recall some of the well-known results from the theory (for theproof, see [8, Proposition 5.8.7]). Lemma 4.4.
Let q ” . Let L i “ K p z i q ( i “ , ) be two cyclic extensions of K of degree 3,given by generating equations z i “ a i . The following statements are equivalent:(1) L “ L .(2) z “ z j c for all ď j ď and c P K .(3) a “ a j c for all ď j ď and c P K . The next Theorem gives ramification data, genus and Galois action for Kummer extensions [7,Proposition 3.7.3].
Theorem 4.5.
Let K { F q be an algebraic function field of characteristic p ą with q ” .Suppose that a P K is an element which satisfies the condition a ‰ w , f or all w P K . Let L “ K p y q with y “ a , so that L { K is a Kummer extension. We then have:(1) L { K is a cyclic Galois extension of degree . The automorphisms of L { K are given by σ p y q “ ξ y ,with ξ a primitive rd root of unity.(2) A place p of K is ramified in L { K if, and only if, p v p p a q , q “ . For a place p of K ramifiedin L , denote by P the unique place of L lying over p . Then the differential exponent d p P | p q isgiven by d p P | p q “ . (3) If at least one place Q satisfies v Q p a q ą , then F q is algebraically closed in L , and g L “ g K ´ ` ÿ p v p p a q , q“ deg p p q , where g L (resp. g K ) is the genus of L { F q (resp. K { F q ). We recall when an Kummer extension is constant.
Theorem 4.6.
Suppose that q ” mod , and that L { F q p x q is cubic and Galois, it is thus a Kummerextension and there is a primitive element y of L { F q p x q having irreducible polynomial T p X q “ X ´ b .Then, L { F q p x q is a constant extension (whence L “ F q p x q ) if, and only if, b “ u β with u P F q , u isnot a cube and β P F q p x q .Proof. If b “ u β with u P F q where u P F q is not a cube and β P F q p x q , then z “ y { β generates L { K and z “ u , by definition of the constant field and the generating equation z “ u (Note that if u was a cube then X ` u would not be irreducible), L is obtained by adjoining constants to F q p x q , and L { F q p x q is constant. Conversely, suppose that b ‰ u β with u P F q where u is not a cube and β P F q p x q .Then, as the polynomial is irreducible, there exists a place p of F q p x q such that p v p p b q , q “ . ByKummer theory [8, Theorem 5.8.12], it follows that p is (fully) ramified in L . As constant extensionsare unramified (see for example [8, Theorem 6.1.3]), we find that L { F q p x q cannot be constant. (cid:3) We finish this section by giving an integral basis for Kummer extensions ([4, Theorem 3]).
Theorem 4.7.
Let L be an Kummer extension of F q p x q , and let y “ a , where the factorisation of a is given by a “ l ź i “ P λ i i , COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 15 where P i P F q r x s , ď λ i ď for each i “ , . . . , l . (This is known to exist; see [8, Example 5.8.9] .)Then t , yS , y S u is an integral basis of L over F q p x q , where for each j “ , , S j “ l ź i “ P r i , j i , with r i , j “ Y j λ i ] , where t x u denotes the integral part of x . Extensions with generating equation y ´ y ´ a “ The Galois criterion.
In the next Theorem, we investigate when a cubic extension with gener-ating equation y ´ y ´ a “ is Galois, similarly to Theorem 3.1 and Corollary 4.2 for Artin-Schreierand Kummer extensions. Theorem 5.1.
Let q ” ´ . Then, a cubic extension L { F q p x q is Galois if, and only if,(a) If p ‰ , L { F q p x q has a primitive element z with minimal polynomial of the form T p X q “ X ´ X ´ b , where b “ PQ , for some P , Q P F q r x s such that p P , Q q “ , and " P “ p A ´ ´ B q Q “ A ` ´ B for some A , B P F q r x s , p A , B q “ .(b) If p “ , there exists a primitive element z of L { F q p x q with minimal polynomial T p X q “ X ´ X ´ b , where b “ PQ , for some P , Q P F q r x s such that p P , Q q “ , and " P “ A Q “ A ` AB ` B , for some A and B P F q r x s and p A , B q “ .Proof. Let q ” ´ .(a) Suppose that L { F q p x q is Galois and p ‰ . By Corollaries 1.2 and 4.3, there exists a primitiveelement z with minimal polynomial of the form T p X q “ X ´ X ´ b . It follows that the discriminantof L { F q p x q satisfies d L { F q p x q “ ´ p´ ` b q . As in the statement of the Theorem, we write b “ P { Q with P , Q P F q r x s and gcd p P , Q q “ . By Lemma 2.1, L { F q p x q is Galois if, and only if, there exists R P F q r x s such that R “ ´ p´ Q ` P q “ ´ p´ Q ` P qp Q ` P q . The polynomials ´ Q ` P and Q ` P are relatively prime; indeed, for if f divides ´ Q ` P and Q ` P , then f divides both Q “ p Q ` P q ´ p´ Q ` P q and P “ p Q ` P q ` p´ Q ` P q ,contradicting gcd p P , Q q “ . Thus, by unique factorisation in F q r x s , it follows that, up to elementsof F ˚ q , Q ` P and ´ Q ` P are squares in F q r x s . Therefore, there exist c , d P F ˚ q with cd equal to ´ ´ up to a square in F ˚ q and A , B P F q r x s such that Q ` P “ cA and ´ Q ` P “ dB . Thus P “ cA ` dB and Q “ cA ´ dB , and Q “ cA ´ dB “ c ´ p c A ´ cdB q “ c ´ pp cA q ` ´ p ´ B q q . Therefore, " cQ “ A ` ´ B cP “ A ´ ´ B , where A “ cA and B “ ´ B . Given A , B as before, for any c P F q , b takes the same value b “ A ´ ´ B A ` ´ B s . Thus, without loss of generality, we have " Q “ A ` ´ B P “ p A ´ ´ B q , for some A and B P F q r x s with p A , B q “ .Conversely, suppose that " Q “ A ` ´ B P “ p A ´ ´ B q for some A and B in F q r x s , and p A , B q “ . Then d L { F q p x q “ ´ p´ ` b q“ ´ ˜ ´ ` ˆ A ´ ´ B A ` ´ B ˙ ¸ “ ´ ˆ ˜ ´p A ` ´ B q ` p A ´ ´ B q p A ` ´ B q ¸ “ ´ ˆ ˆ ´ ˆ ´ A B p A ` ´ B q ˙ “ ˆ A B p A ` ´ B q ˙ , whence d L { F q p x q is a square. By Lemma 2.1, it follows that L { F q p x q is Galois.(b) Suppose that L { F q p x q is Galois, and that p “ . By Lemma 1.3 and Corollary 4.3, there is aprimitive element z of L { F q p x q with minimal polynomial of the form T p X q “ X ´ X ´ b . ByLemma 2.3, we know that L { F q p x q is Galois if, and only if, the resolvent polynomial R p X q “ X ` bX ` p´ ` b q of T p X q is reducible. This is the same as requiring that the polynomial X ` X “ { b ´ isreducible, which (as this is a polynomial of degree 2) is equivalent to the existence of at least one w P K such that w ´ w “ { b ´ “ { b ` . As p “ , we have p w ` { b q ´ p w ` { b q “ ` { b .The latter is equivalent to the existence of w P F q p x q such that w ´ w “ b ` b . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 17
We write b “ PQ where P , Q P F q r x s with p P , Q q “ and w “ CD where C , D P F q r x s with p C , D q “ . We thus find that C D ´ CD “ P ` QP and C ´ CDD “ P ` QP . As p C ´ CD , D q “ and p P ` Q , P q “ , it follows that up to a constant not affecting the valueof b , " P “ D P ` Q “ C ´ CD ; equivalently, " P “ D Q “ C ´ CD ` D . Conversely, suppose b “ PQ , where " P “ D Q “ C ´ CD ` D where C , D P F q r x s with p C , D q “ . Then the polynomial X ´ X “ b ` b has CD as root and theTheorem follows. (cid:3) Remark 5.2.
We note that under the assumptions and notation of the previous Theorem, Q isconstant if, and only if, b is constant. Indeed, if Q is constant, then with the notation of the previousTheorem, we write Q “ A ` ´ B . Letting A “ a m x m ` a m ´ x m ´ ` ¨ ¨ ¨ ` a and B “ b n x n ` b n ´ x n ´ ` ¨ ¨ ¨ ` b , it follows from the fact that Q is constant that if either of m or n is positive, then m “ n . This impliesthat a n ` ´ b n “ , and hence that p´ q ´ “ p b n { a n q where b n { a n P F q , contradicting that q ” ´ .More precisely, we have the following Theorem, which characterises the Galois extensions of theform y ´ y “ a in terms of the factorisation of the denominator of a . Theorem 5.3.
Let q ” ´ . Suppose that L { F q p x q is a Galois cubic extension. By Theorem1.2, L { F q p x q has a primitive element z with minimal polynomial of the form T p X q “ X ´ X ´ b . Write b “ PQ with P , Q P F q r x s such that p P , Q q “ . Then, Q “ w ź i Q e i i with Q i unitary irreducible of even degree, w P F ˚ q , and e i a positive integer.Proof. By [3, Theorem 3.46], an irreducible polynomial over F q of degree n factors over F q k r x s into gcd p k , n q irreducible polynomials of degree n { gcd p k , n q . It follows that an irreducible polynomial over F q factors in F q into polynomials of smaller degree if, and only if, gcd p , n q ą , i.e., | n . We notethat the rings F q r x s and F q r x s “ F q p u qr x s for some u P F q z F q are both unique factorisation domains.Also, the irreducible polynomials in F q r x s are those irreducible polynomials of odd degree in F q r x s orthe irreducible polynomials occurring as factors of irreducible polynomials of even degree in F q r x s . (1) If p ‰ , let t be a root of x ` ´ in F q . From the previous Theorem, we know that, Q “ A ` ´ B , where A , B P F q r x s and p A , B q “ . We consider the norm map N : F q r x s Ñ F q r x s α ` t β ÞÑ α ` ´ β “ p α ` t β qp α ´ t β q “ p α ` t β q q ` . As, the element ´ t “ t q is the other root of the polynomial X ` ´ “ in F q . Indeed, thecoefficients of S p X q “ X ` ´ are in F q , whence S p t q q “ S p t q q “ , and t ‰ t q , as t R F q .Clearly, the map N is multiplicative, and it follows that a polynomial Q “ w ś i Q e i i is ofthe form N p A ` tB q “ Q with A , B P F q r x s coprime if, and only if, each Q i is of the form N p A i ` tB i q “ Q i , with A i , B i P F q r x s coprime.Indeed, observe that ˝ If U is an irreducible polynomial of odd degree in F q r x s , then N p U q “ Q i , ˝ If U is an irreducible polynomial occurring as a factor of an irreducible polynomial of evendegree in F q r x s , then it is of the form U “ A ` tB with A , B P F q r x s , B ‰ and p A , B q “ .Then N p U q “ A ` p´ q ´ B , Thus, N p A i ` tB i q “ Q i , with A i , B i P F q r x s coprime if only if A i ` tB i is an irreducible polynomialoccurring as a factor of an irreducible polynomial of even degree in F q r x s . We therefore concludethe Theorem if p ‰ .(2) If p “ , similarly, let ξ be a primitive rd root of unity in F q , whence ξ ` ξ ` “ . Fromthe previous Theorem, we know that, Q “ A ` AB ` B , where A , B P F q r x s and p A , B q “ . We consider now the norm map N : F q r x s Ñ F q r x s α ` ξβ ÞÑ α ` αβ ` β “ p α ` ξβ qp α ` ξ β q “ p α ` ξβ q q ` Indeed, ξ “ ξ q , as it is a root distinct from ξ of the polynomial X ` X ` , which hascoefficients in F q . As before, the map N is multiplicative. We thus deduce the Theorem as inthe previous case. (cid:3) Any Q of the form given in the previous Theorem is realisable as the denominator of b . Here, wegive a recipe to construct Galois cubic extensions without primitive rd roots of unity with a given Q as in the previous Theorem. We begin with the following Lemma. Lemma 5.4.
Suppose q ” ´ . Given w P F q , there are q ` ways to write w as(1) w “ u ` p´ q ´ v , for some u , v P F q , if p ‰ ;(2) w “ u ` uv ` v , for some u , v P F q , if p “ . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 19
Proof. (1) If p ‰ , as in the previous Theorem, let t be one of the solutions of the equation X ` ´ “ in F q . As, ´ t “ t q , we have, w “ p u ` tv qp u ´ tv q “ p u ` tv qp u ` t q v q “ p u ` tv q q ` and “ w q ´ “ p u ` tv q q ´ . If w “ then u ` tv is a q ` th root of unity and there are q ` of those in F q . Otherwise, u ` tv is a q ´ st root of unity, say u ` tv “ ζ m where ζ be a primitive q ´ st root of unityand m a positive integer. Also, w “ ζ l for some l positive integer; moreover, w q ´ “ , whence m p q ´ q is divisible by q ´ . It follows that l “ s p q ` q for some positive integer s , and m “ s , s ` p q ´ q , . . . , s ` q p q ´ q give q ` distinct u ` tv “ ζ m . Hence q ` distinct waysto write w as w “ N p u ` tv q “ u ` ´ v .(2) If p “ , let ξ be one of the solutions of the equation X ` X ` “ in F q . Then ξ “ ξ ` “ ξ q is also the other root of the polynomial X ` X ` thus the argument developed for p ‰ canbe applied in this case too. (cid:3) Lemma 5.5.
Let Q “ w r ź i “ Q e i i with Q i unitary irreducible in F q r x s of even degree, w P F ˚ q , and e i a positive integer. Then there areat most r ` p q ` q , p A , B q P F q r x s ˆ F q r x s relatively prime such that(1) Q “ A ` p´ q ´ B , for some A , B P F q , if p ‰ ;(2) Q “ A ` AB ` B , for some A , B P F q , if p “ .Proof. The degree of Q i is even, whence by [3, Theorem 3.46], each Q i factors in F q p t qr x s into gcd p , n q “ irreducible polynomials of degree n { gcd p , n q “ n { .(1) Suppose that p ‰ . We have that Q i “ p A i ` tB i qp A i ´ tB i q “ A i ` ´ B i , The factors of Q i are then p A i ` tB i q and p A i ´ tB i q , which are unique up to a unit. Also, as Q i is irreducible in F q r x s , we have that A i and B i are coprime. Suppose that Q i “ N p C i ` tD i q “ N p A i ` tB i q . As F q r x s is a unique factorisation domain, we have C i ` tD i “ a i p A i ˘ tB i q and C i ´ tD i “ b i p A i ¯ tB i q for some a i , b i P F q “ F q p t q , a i b i “ with A i and B i P F q r x s . We write a i “ u i ` tv i and b i “ w i ` tz i where u i , v i , w i , z i P F q . As a i b i “ p u i w i ´ ´ v i z i q ` t p v i w i ` u i z i q “ , it followsthat " “ u i w i ´ ´ v i z i “ v i w i ` u i z i Thus, " C i ` tD i “ p u i ` tv i qp A i ˘ tB i q “ u i A i ¯ ´ v i B i ` p A i v i ˘ B i u i q tC i ´ tD i “ p w i ` tz i qp A i ¯ tB i q “ w i A ˘ ´ z i B ` p Az i ¯ Bw i q t As a consequence, " C i “ u i A i ¯ ´ v i B i and D i “ A i v i ˘ B i u i C i “ wA i ˘ ´ zB i and D i “ ´p A i z i ¯ B i w i q Subtracting C i “ u i A i ¯ ´ v i B i and C i “ w i A i ˘ ´ z i B i , we find p u i ´ w i q A i “ ¯ ´ p v i ` z i q B i As p A i , B i q “ , u i “ w i and v i “ ´ z i , whence “ u i ` ´ v i . Thus, we can write Q e i i as an element in the image of the norm map N , that is, Q e i i “ N ˜ e i ź l “ p u i , l ` tv i , l q e i ź l “ p A i ` ǫ i , l tB i q ¸ , where u i and v i P F q satisfy “ u i , l ` ´ v i , l and ǫ i , l P t˘ u . Moreover, if in this productwe have at least one ǫ i , l “ and at least one ǫ i , l “ ´ for some l ‰ l , then it would leadto A and B sharing a divisor, which violates that they are coprime. It follows that the onlypossibility is to write either Q e i i “ N ˜˜ e i ź l “ p u i , l ` tv i , l q ¸ p A i ` tB i q e i ¸ or Q e i i “ N ˜˜ e i ź l “ p u l , i ` tv l , i q ¸ p A i ´ tB i q e i ¸ By the previous Lemma, we write w as w “ N p µ ` t ν q for some µ and ν P F q , and Q “ N ˜ p µ ` t ν q r ź i “ ˜ e i ź l “ p u i , l ` tv i , l q ¸ p A i ` ǫ i tB i q e i ¸ , where ǫ i “ ˘ with µ, ν, u i , l , v i , l P F q . We have p µ ` t ν q ś ri “ ś e i l “ p u i , l ` tv i , l qq “ α ` t β for some α, β P F q and w “ N p α ` t β q . By Lemma 5.4, there exist q ` such α and β . Furthermore, A ` tB “ p α ` t β q r ź i “ p A i ` ǫ i tB i q e i . The result follows.(2) Suppose that p “ . We have that Q i “ p A i ` ξ B i qp A i ` ξ B i q “ A i ` A i B i ` B i , The factors of Q i are then p A i ` ξ B i q and p A i ` ξ B i q unique up to a unit. Also, as Q i is irreduciblein F q , we have that A i and B i are coprime. Suppose that Q i “ N p C i ` ξ D i q “ N p A i ` ξ B i q . As F q r x s is a unique factorisation domain, we have C i ` ξ D i “ a i p A i ` ξ B i q and C i ` ξ D i “ b i p A i ` ξ B i q or C i ` ξ D i “ a i p A i ` ξ B i q and C i ` ξ D i “ b i p A i ` ξ B i q for some a i , b i P F q “ F q p t q , such that a i b i “ , as Q i is supposed unitary. We write a i “ u i ` ξ v i , b i “ w i ` ξ z i where u i , v i , w i , z i P F q . As a i b i “ p u i w i ` z i v i q ` ξ p u i z i ` v i w i ` v i z i q “ then " “ u i w i ` z i v i “ u i z i ` v i w i ` v i z i COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 21 (a)
Case 1: C i ` ξ D i “ a i p A i ` ξ B i q “ p u i A i ` v i B i ` u i B i q ` p v i A i ` u i B i q ξ and C i ` ξ D i “ b i p A i ` ξ B i q “ p w i A i ` z i B i q ` p z i B i ` z i A i ` w i B i q ξ Thus $’’&’’% C i “ u i A i ` v i B i ` u i B i D i “ v i A i ` u i B i C i “ w i A i ` z i A i ` w i B i D i “ z i B i ` z i A i ` w i B i Hence, " p u i ` w i ` z i q A i “ p v i ` u i ` w i q B i p v i ` z i q A i “ p u i ` z i ` w i q B i Then, as p A i , B q “ , $’’&’’% v i ` z i “ u i ` z i ` w i “ u i w i ` z i v i “ u i z i ` v i w i ` v i z i “ Hence, $&% v i ` z i “ u i ` z i ` w i “ N p u i ` ξ v i q “ u i ` u i v i ` v i “ (b) Case 2: C i ` ξ D i “ a i p A i ` ξ B i q “ p u i A i ` v i B i q ` p v i A i ` u i B i ` v i B i q ξ and C i ` ξ D i “ b i p A i ` ξ B i q “ p w i A i ` z i B i ` w i B i q ` p z i A i ` w i B i q ξ Thus $’’&’’% C i “ u i A i ` v i B i D i “ v i A i ` u i B i ` v i B i C i “ w i A i ` z i A i ` z i B i D i “ z i A i ` w i B i Hence, " p u i ` w i ` z i q A i “ p v i ` z i q B i p v i ` z i q A i “ p u i ` v i ` w i q B i Thus, as before $&% v i ` z i “ u i ` z i ` w i “ N p u i ` ξ v i q “ u i ` u i v i ` v i “ As a consequence, we can always write Q e i i as an element in the image of the norm map N ,whence Q e i i “ N ˜ e i ź l “ p u i , l ` ξ v i , l q e i ź l “ θ i , l ¸ , where u i , l and v i , l P F q satisfies “ u i , l ` u i , l v i , l ` v i , l where θ i , l P t A i ` ξ B i , A i ` ξ B i u . Moreover,if in this product we have at least one θ i , l “ A i ` ξ B i and at least one θ i , l “ A i ` ξ B i for some l ‰ l , then it would lead to A and B that would not be coprime. It follows that theonly possibility is to write either Q e i i “ N ˜˜ e i ź l “ p u i , l ` ξ v i , l q ¸ p A i ` ξ B i q e i ¸ or Q e i i “ N ˜˜ e i ź l “ p u l , i ` ξ v l , i q ¸ p A i ` ξ B i q e i ¸ By Lemma 5.4, we write w as w “ N p µ ` ξν q for some µ and ν P F q , and Q “ N ˜ p µ ` ξν q r ź i “ ˜ e i ź l “ p u i , l ` ξ v i , l q ¸ θ e i i ¸ , where ǫ i “ ˘ with µ, ν, u i , l , v i , l P F q where θ i P t A i ` ξ B i , A i ` ξ B i u . We have p µ ` ξν q ś ri “ ś e i l “ p u i , l ` ξ v i , l qq “ α ` ξβ for some α, β P F q and w “ N p α ` ξβ q , and by Lemma 5.4,there are again q ` such α and β . Finally, A ` ξ B “ p α ` ξβ q r ź i “ θ e i i , which yields the result in this case. (cid:3) The irreducibility criterion.
In the following Theorem and Corollary, we see that if L { K isa Galois cubic extension with a generating equation of the form y ´ y “ a where a “ PQ and P , Q P F q r x s with p P , Q q “ , then Q cannot be a cube, up to a unit. Theorem 5.6.
Let q ” ´ . Let P p X q “ X ´ X ´ a be a polynomial, where a “ PQ , P , Q P F q r x s ,and p P , Q q “ .(1) If p ‰ , suppose that P “ p A ´ ´ B q and Q “ A ` ´ B , for some A , B P F q r x s with p A , B q “ . Let t P F q be a root of the polynomial X ` ´ . Then T p X q is a reducible polynomialover F q p x q if, and only if, p A ` tB q is a cube in F q r x s .(2) If p “ , suppose that P “ A and Q “ A ` AB ` B , for some A , B P F q r x s with p A , B q “ .Let ξ P F q be a primitive third root of unity. Then T p X q is a reducible polynomial over F q p x q if, and only if, p B ` ξ A q is a cube in F q r x s .Proof. First, if T p X q is reducible over F q p x q then there is fg with f , g P F q r x s and p f , g q “ such that T ˆ fg ˙ “ That implies that ˆ fg ˙ ´ ˆ fg ˙ ´ b “ That is f ´ f g g “ PQ As p f ´ f g , g q “ and p P , Q q “ then Q “ g up to a constant.Suppose that Q is a cube up to a unit c P F ˚ q . Then up to multiplication by c , without loss ofgenerality, we can suppose that Q is a cube say Q “ g with F q p x q . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 23 (1) If p ‰ , let t be a root of x ` ´ in F q . Suppose L { F q p x q has a generating equation of the form y ´ y “ a where a “ PQ where P “ p A ´ ´ B q and Q “ A ` ´ B , for some A , B P F q r x s and p A , B q “ . By Lemma 5.5, A ` tB “ p u ` tv qp α ˘ t β q and A ´ tB “ p u ´ tv qp α ¯ t β q , for some u , v P F q and α , β P F q p t q with p α, β q “ , as A ` ´ B “ N p A ` tB q “ p A ` tB qp A ´ tB q “ Q “ g .We now examine the polynomial X ´ X ´ a over F q p t qp x q . We have a ´ “ ˆ t ABA ` ´ B ˙ “ δ , where δ “ t ABA ` ´ B . By the proof of Theorem 4.1, we have that as a ´ is a square over F q p t q , whence F q p t qp x q L { F q p t qp x q is purely cubic, so that there is w “ y ` ky ´ such that w “ b , where k “ ´ a ˘ δ “ ´ A ` ´ B ˘ tABA ` ´ B “ ´ A ` ´ B ˘ tABA ` ´ B “ ´p A ¯ tB q Q , and furthermore, b “ δ k . As we supposed Q “ A ` ´ B to be a cube and A ` tB “ p u ` tv qp α ˘ t β q , then we have that X ´ b is reducible over F q p t qp x q if, and only if, X ´ X ´ a is reducible over F q p t qp x q , whichis true if, and only if, A ` tB is a cube in F q p t qp x q . (This follows from the fact that A ` tB isa cube if, and only if, A ´ tB is a cube, as p A ` tB qp A ´ tB q “ A ` ´ B “ Q “ g .) Also, X ´ X ´ a is irreducible over F q p t qp x q implies X ´ X ´ a is irreducible over F q p x q . Thus, if T p X q is reducible over F q p x q , then p A ` tB q is a cube in F q p x q .For the converse, suppose that Q “ A ` ´ B and A ` tB is a cube in F q p t qr x s , say, A ` tB “ p α ` t β q . Then A ` tB “ p α ` t β q “ β p α ` p´ q ´ β q t ` α p α ´ β q and A “ β p α ` p´ q ´ β q and B “ α p α ´ β q . We now let r “ p α ´ ´ β q α ` ´ β P F q p x q . We observe that r ´ r “ p α ´ α β q ´ ´ p α β ´ ´ β q p α ´ α β q ` ´ p α β ´ ´ β q “ PQ . It follows that T p X q is reducible.(2) If p “ , let ξ be a primitive rd root of unity in F q . Suppose that L { K has a generating equationof the form y ´ y “ a , where a “ PQ , P “ A , and Q “ A ` AB ` B for some A and B P F q r x s and p A , B q “ . From Lemma 5.5, A ` ξ B “ p u ` ξ v qp α ` ξβ q and A ` ξ B “ p u ` ξ v qp α ` ξ β q ,as A ` AB ` B “ N p A ` ξ B q “ p A ` ξ B qp A ` ξ B q “ Q , for some u , v P F q and α , β P F q p t q with p α, β q “ . We examine the polynomial X ´ X ´ a over F q p ξ qp x q . We have a “ Z W p Z ` W q for W “ ξ A ` B , Z “ A and p W , Z q “ as p A , B q “ . By Theorem 4.1, w “ y ` ky is suchthat w “ b , where k “ W p Z ` W q Z and b “ W p Z ` W q P “ ˆ WP ˙ W p ξ A ` B q “ ˆ WP ˙ Q p ξ A ` B q . As Q is a cube by supposition and A ` ξ B “ p u ` ξ v qp α ` ξβ q for some u , v P F q and α , β P F q p t q with p α, β q “ , we have X ´ b is reducible over F q p t qp x q if, and only if, X ´ X ´ a is reducibleover F q p t qp x q , which is true if, and only if, B ` ξ A is a cube in F q p t qp x q . Also, irreducibility of X ´ X ´ a over F q p ξ qp x q implies that X ´ X ´ a is irreducible over F q p x q . Thus, if T p X q isreducible over F q p x q , then p B ` ξ A q is a cube in F q p x q . (This follows from the fact that B ` ξ A is a cube if, and only if, B ` ξ A is a cube, as p B ` ξ A qp B ` ξ A q “ A ` AB ` B “ Q “ g .)Conversely, suppose that B ` ξ A is a cube in F q p ξ q , say, p B ` ξ A q “ p β ` ξα q . Then p B ` ξ A q “p β ` ξα q “ α ξ ` βα ξ ` β αξ ` β “ α ` βα p ξ ` q ` β αξ ` β “ α ` βα ` β ` βα p α ` β q ξ, whence B “ α ` βα ` β and A “ βα p α ` β q . We now let r “ α ` β β ` βα ` α P F q p x q . We then observe that, for this choice of r , r ´ r “ PQ , whence T p X q is reducible. (cid:3) Corollary 5.7.
Suppose that q ” ´ mod , and that L { F q p x q is cubic and Galois, so that there existsa primitive element z of L { F q p x q with irreducible polynomial T p X q “ X ´ X ´ b (see Theorem 5.1).(1) If p ‰ , then b “ PQ where P “ p A ´ ´ B q and Q “ A ` ´ B , for some A , B P F q r x s with p A , B q “ . The extension L { F q p x q is constant if, and only if, p A ` tB q “ p u ` tv qp α ` t β q , with u , v P F q r x s and u ` tv not a cube in F q , where t P F q is a root of the polynomial X ` ´ .(2) When p “ , b “ PQ where P “ A and Q “ A ` AB ` B , for some A , B P F q r x s with p A , B q “ .The extension L { F q p x q is constant if, and only if, p B ` ξ A q “ p u ` ξ v qp β ` ξα q , with u , v P F q r x s and u ` ξ v not a cube in F q , where ξ P F q is a primitive third root of unity.In either case, if L { F q p x q is constant, then L “ F q p x q . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 25
Proof.
We begin by noting that F q L { F q p x q is constant if, and only if, L { F q p x q is constant. Indeed, if L { F q p x q is constant, then clearly F q L { F q p x q is constant. If L { F q p x q is not constant, then as there isno unramified extension of F q p x q (see [8, Theorem 6.1.3]), there would be a ramified place in F q p x q in L , and then a place of F q p x q above this ramified place would also ramify in F q L , so that F q L { F q p x q would also not be constant.(1) If p ‰ , b “ PQ where P “ p A ´ ´ B q and Q “ A ` ´ B , for some A and B P F q r x s and p A , B q “ . In F q p x q , using the notation of the proof of the previous Theorem, we know thatthere is w a generator of F q L { F q p x q such that w “ b where b “ δ ˆ ´p A ˘ tB q A ` ´ B ˙ “ δ ˆ ´p A ˘ tB q p A ` tB qp A ´ tB q ˙ “ δ ˆ ´ A ˘ tBA ¯ tB . ˙ and taking z “ ´ w δ , we have that z “ A ˘ tBA ¯ tB . As F q p ξ qr x s is a unique factorisation domain and p A ` tB q and p A ´ tB q are coprime, we know that A ˘ tBA ¯ tB is a cube if, and only if, p A ` tB q and p A ´ tB q are a cube.Also, p A ` tB q is a cube if, and only if, A ´ tB is a cube, indeed p A ` tB qp A ´ tB q “ Q and Q P F q r x s , thus each factor of p A ` tB q have their conjugates appearing to the same power.Thus, F q L { F q p x q is constant if, and only if, A ` tB is a cube up to a non-cube constant in F q (see Lemma 4.6).(2) If p “ , b “ PQ where P “ A and Q “ A ` AB ` B , for some A and B P F q r x s and p A , B q “ .In F q p x q , using the notation of the proof of the previous Theorem, we know that there is w agenerator of F q L { F q p x q such that w “ b where b “ ˆ WP ˙ p ξ A ` B qp ξ A ` B qp ξ A ` B q “ ˆ WP ˙ p ξ A ` B q p ξ A ` B q and taking z “ PwW , then we obtain z “ p ξ A ` B q p ξ A ` B q . As F q p ξ qr x s is a unique factorisationdomain and p ξ A ` B q and p ξ A ` B q are coprime, we know that p ξ A ` B q p ξ A ` B q is a cubeif, and only if, p ξ A ` B q and p ξ A ` B q are a cube. Note that p ξ A ` B q is a cube if, and onlyif, ξ A ` B is a cube, indeed p ξ A ` B qp ξ A ` B q “ Q and Q P F q r x s , thus each factor of ξ A ` B have their conjugates appearing to the same power. Thus, F q L { F q p x q is constant if, and onlyif, ξ A ` B is a cube up to a non-cube constant in F q (once again, we refer the reader to Lemma4.6). (cid:3) The Galois action on the generator.
The following Theorem describes the Galois action onthe generator z of a cubic Galois extension satisfying a minimal equation of the form X ´ X ´ b “ .This is not as simple as for the Artin-Schreier or Kummer generators (see Theorems 3.3 and 4.5), butmay be concisely described, which we now do here. We note that in the following Theorem, we do notassume that K “ F q p x q . Theorem 5.8.
Suppose q ” ´ . Let L { K be a Galois cubic extension, and let z be the elementof Corollary 1.4 which has minimal polynomial of the form T p X q “ X ´ X ´ a . Then, the Galoisaction of L { K on z is given in the following way: σ p z q “ ´ ` fa z ` f z ` p ` f q a and σ p z q “ ` fa z ` p´ ´ f q z ´ p ` f q a for σ P Gal p L { K q , where f is one root of the polynomial S p X q “ ˆ ´ a ˙ X ` ˆ ´ a ˙ X ` ˆ ´ a ˙ . (Note that ´ ´ f is the other root of S p X q .) Moreover,(a) If p ‰ , then f “ ´ ` a ´ ˘ ˘ a δ p a ´ q where δ “ D , with D “ ´ p a ´ q is the discriminant of the polynomial P p X q “ X ´ X ´ a .(b) If p “ , a f is a root of the resolvent polynomial R p X q “ X ` aX ` p ` a q of T p X q .Proof. Suppose that q ” ´ . Let σ P Gal p L { K q . As σ p z q P L , we let σ p z q “ ez ` f z ` g , with e , f , g P K . Then σ p z q“ e p ez ` f z ` g q ` f p ez ` f z ` g q ` g “ e z ` e f z ` eg ` e f z ` e gz ` e f gz ` f ez ` f z ` f g ` g “ e z ` e az ` e f z ` eg ` e f z ` e f a ` e gz ` e f gz ` f ez ` f z ` f g ` g “p e ` e f ` f e ` e g q z ` p e a ` f ` e f ` e f g q z ` p eg ` f g ` g ` e f a q . Moreover, as z , σ p z q and σ p z q are the three roots of P p X q and P p X q “ X ´ X ´ a , we have Tr p z q “ z ` σ p z q ` σ p z q “ . That is, p e ` e f ` f e ` e g ` e q z ` p e a ` f ` e f ` e f g ` f ` q z ` p eg ` f g ` g ` e f a ` g q “ . As , z , z is a basis for L over K , we obtain the system $&% e ` e f ` f e ` e g ` e “ e a ` f ` e f ` e f g ` f ` “ eg ` f g ` g ` e f a ` g “ We note that e ‰ , as K does not contain primitive rd roots of unity, using the second equation ofthe system. Thus, the previous system simplifies to $&% e ` f ` f ` eg ` “ e a ` f ` e f ` e f g ` f ` “ eg ` f g ` g ` e f a ` g “ (I)We denote z : “ σ p z q , then z : “ σ p z q “ ´ z ´ z . As the linear coefficient of P p X q is equal to ´ , wehave that ´ “ zz ` zz ` z z “ zz ` z p´ z ´ z q ` z p´ z ´ z q“ ´ z ´ z ´ z z COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 27 “ ´ z ´ p ez ` f z ` g q ´ z p ez ` f z ` g q“ ´ z ´ e z ´ f z ´ g ´ e f z ´ egz ´ f gz ´ ez ´ f z ´ gz “ ´ z ´ e z ´ ae z ´ f z ´ g ´ e f z ´ e f a ´ egz ´ f gz ´ ez ´ ea ´ f z ´ gz “ ´p ` e ` f ` eg ` f q z ´ p ae ` e f ` f g ` e ` g q z ´ p g ` ea ` e f a q . We therefore obtain the following system: $&% ` e ` f ` eg ` f “ ae ` e f ` f g ` e ` g “ g ` ea ` e f a ´ “ (II)As the constant coefficient is equal to ´ a , we must have a “ zz z “ zz p´ z ´ z q “ ´ zz ´ z z “ ´r z p ez ` f z ` g q ` z p ez ` f z ` g qs“ ´p e z ` f z ` g z ` e f z ` egz ` f gz ` ez ` f z ` gz q“ ´p e z ` e az ` f z ` f a ` g z ` e f z ` e f az ` egz ` ega ` f gz ` ez ` eaz ` f z ` f a ` gz q“ ´p e z ` e a ` e az ` f z ` f a ` g z ` e f z ` e f az ` egz ` ega ` f gz ` ez ` eaz ` f z ` f a ` gz q“ ´p e a ` e f ` f g ` e ` g q z ´ p e ` f ` g ` e f a ` eg ` ea ` f q z ´ p e a ` f a ` ega ` f a q . This leads $&% e a ` e f ` f g ` e ` g “ e ` f ` g ` e f a ` eg ` ea ` f “ e a ` f a ` ega ` f a ` a “ As a ‰ , the system becomes $&% e a ` e f ` f g ` e ` g “ e ` f ` g ` e f a ` eg ` ea ` f “ e ` f ` eg ` f ` “ (III)Together, the systems (I), (II), (III) yield the system of six equations $’’’’’’&’’’’’’% e ` f ` f ` eg ` “ p q e a ` f ` e f ` e f g ` f ` “ p q eg ` f g ` g ` e f a “ p q ae ` e f ` f g ` e ` g “ p q g ` ea ` e f a ´ “ p q e ` f ` g ` e f a ` eg ` ea ` f “ p q Subtracting Eq. (2) from (1) then yields “ e ` f ` f ` eg ` ´ p e a ` f ` e f ` e f g ` f ` q “ e ` ge ´ e a ´ e f ´ f ge . Thus, as e ‰ (because K does not contain primitive rd roots of unity), division by e yields e ` g ´ e a ´ e f ´ f g “ . p q The sum of Eqs. p q and p q implies e ` g “ . As p ‰ , it follows that g “ ´ e . p˚q By substitution of this identity in Eq. p q , we obtain ae ` e f ` e “ ae ` e f ´ e f ` e ´ e “ . As e ‰ and a ‰ , it follows that ae ` f ` “ , and hence that e “ ´ ` fa . p˚˚q Eq. (1) then reads as “ e ` f ` f ´ e ` “ ´ e ` f ` f ` “ ´ ˆ ´ ` fa ˙ ` f ` f ` “ ´ p ` f ` f q a ` f ` f ` “ ˆ ´ a ˙ f ` ˆ ´ a ˙ f ` ˆ ´ a ˙ ˝ If p ‰ , the discriminant of the polynomial S p X q : “ ˆ ´ a ˙ X ` ˆ ´ a ˙ X ` ˆ ´ a ˙ is equal to ∆ “ ˆ ´ a ˙ ´ ˆ ´ a ˙ ˆ ´ a ˙ “ ˆ ´ a ˙ ˆˆ ´ a ˙ ´ ˆ ´ a ˙˙ “ ´ ˆ ´ a ˙ “ a D , where D “ ´ p a ´ q is the discriminant of the polynomial X ´ X ´ a , which is a squareas L { K is a Galois. Thus ∆ is a square. Thus S p X q has a root and f can be chosen as one ofthese roots. The latter together with p˚q and p˚˚q give part p a q of the Theorem. ˝ If p “ , by definition, the resolvent R p X q of T p X q is equal to R p X q “ X ` aX ` ` a and has a root that we denote γ , as L { K is Galois. We observe that S p X q “ R p X q{ a “ Y ` Y ` ` { a where Y “ X { a . Thus, γ a is a root of S p X q . The later together with p˚q and p˚˚q give part p b q of the Theorem.Lastly, we give a brief verification that e , g , and f as defined satisfy the equations p q ´ p q . Equations p q and p q have already been verified before. Via the substitution g “ ´ e , Eq. p q becomes “ e a ` f ` e f ` e f g ` f ` “ e a ` e f ` e f g ` e , COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 29 which becomes p q by division of e . Via the substitutions g “ ´ e and e “ ´ ` fa , Eq. p q becomes eg ` f g ` g ` e f a “ e ´ f e ´ e ` e f a “ e p e ´ f ´ ` e f a q“ e p e ´ f ´ ´ f ´ f q“ ´ e p´ e ` ` f ` f q “ , which is satisfied as Eq. p q is satisfied. Once again via substitution with g “ ´ e and e “ ´ ` fa ,Eq. p q becomes “ e ` ea ` e f a ´ “ e ´ f ´ f ´ “ ´ p´ e ` f ` f ` q , which is satisfied as Eq. p q is satisfied. Finally, via substitution with g “ ´ e and f ` f “ e ´ ,Eq. p q becomes “ e ` f ` e ` e f ´ e ` ea ` f “ e ´ ` ea ` e f a , which is satisfied as p q is satisfied. (cid:3) Generators with minimal equation X ´ X ´ a “ . The following Theorem gives the formof any generator with minimal equation X ´ X ´ a “ for some a P K . This result is equivalent toLemma 3.2 and 4.4. In particular, we discover that there are infinitely many of those for a given Galoisextension of this form. We give the proof in the Appendix for clarity and brevity, as the argument isquite long. Theorem 5.9.
Suppose that q ” ´ . Let L i “ K p z i q{ K ( i “ , ) be two cyclic extensions ofdegree such that z i ´ z i “ a i P K . The following are equivalent:(1) L “ L ;(2) z “ φ z ` χ z ´ φ , where φ, χ P K satisfy the equation χ ` a φχ ` φ “ . Moreover, if K “ F q p x q , when the above conditions are satisfied, then:(a) If p ‰ , there are relatively prime polynomials C , D P F q r x s φ “ CDQ δ p D ` ´ C q and χ “ ´ P CD ˘ δ p C ´ ´ D q δ p D ` ´ C q , where δ P K is such that δ “ Q d L { F q p x q and a “ P { Q , P , Q P F q r x s with p P , Q q “ and a “ p´ ` a q φ ` a φ χ ` φχ ` a χ . (b) If p “ ,(i) if φ “ , then χ “ ˘ . If χ “ , then a “ a , and if χ “ ´ , then a “ ´ a .(ii) if φ ‰ , then χ {p a φ q is a solution of X ´ X ´ φ ´ a φ “ and a “ a p a φ ` φ χ ` χ q . Ramification and splitting.
The next Theorem studies the ramification for cubic extensionswith a primitive element y whose minimal equation is of the form y ´ y ´ a “ . From it, we obtainthe ramification of the extension via the decomposition of the denominator of a into prime factors.Before doing this, we observe via the following Lemma that the valuation of such an element y at aninfinite place is always nonnegative. Lemma 5.10.
Let p ‰ and q ‰ . Let L { F q p x q be a Galois cubic extension with primitiveelement z , which has minimal polynomial over F q p x q equal to T p X q “ X ´ X ´ a . Then the place p corresponding to the pole divisor of x in F q p x q satisfies v p p b q ě .Proof. By contradiction, suppose that the pole divisor p of x in F q p x q satisfies v p p a q ă . ByTheorem 5.1, with a “ P { Q for relatively prime P , Q , there exist A , B P F q r x s relatively prime suchthat P “ p A ´ ´ B q , and Q “ A ` ´ B . In particular, as v p p a q ă , it follows that deg ˆ r A ´ ´ B s ˙ “ deg p P q ą deg p Q q “ deg ˆ r A ` ´ B s ˙ . Let m “ deg p A q and n “ deg p B q , with A “ ř mi “ a i x i and B “ ř nj “ b j x j . By the previous inequality,we obtain deg p A ` ´ B q ă max t m , n u , which implies that m “ n and a n ` ´ b n “ . Thus p´ q ´ “ p b n { a n q , which is a square in F q , contradicting that q ” ´ . (cid:3) In general, let L { F q p x q be a cubic function field. The discriminant disc x p L q of L over F q r x s is equal tothe discriminant of any basis of the integral closure O L of L over F q r x s . If ω is a generator of L { F q p x q which is integral over F q r x s , the discriminant ∆ p ω q of ω differs from the discriminant disc x p L q by anintegral square divisor, which we denote by I “ ind p ω q ; in particular, we have ∆ p ω q “ I disc x p L q .By definition, we also have p disc x p L qq F q r x s “ pB L { F q p x q q F q r x s , where B L { F q p x q is the discriminant ideal of L { F q p x q [8, Definition 5.6.8]. In the following Theorem, we prove for our extensions that the place p at infinity for x is unramified in L , which in turn implies that p does not appear in B L { F q p x q , whence B L { F q p x q “ pB L { F q p x q q F q r x s . Theorem 5.11.
Suppose q ” ´ . Let L { F q p x q be a Galois cubic geometric extension and y aprimitive element with f p y q “ y ´ y ´ a “ . Then the ramified places of F q p x q in L are preciselythe places p of F q p x q such that v p p a q ă and p v p p a q , q “ .Proof. We first prove that all places p of F q p x q such that v p p a q ă and p v p p a q , q “ are fully ramifiedin L . Suppose that p is a place of F q p x q , v p p a q ă and p v p p a q , q “ . Let P be a place of L which liesabove a place p of K , we denote e p P | p q the ramification index. As v P p a q “ v P p y ´ y q ă , then v P p y q ă . Indeed, if v P p y q “ , then v P p y ´ y q ě v P p y q “ , and if v P p y q ‰ then v P p y q ‰ v P p´ y q , and if v P p y q ą , then v P p y ´ y q “ v P p y q ą . It follows that v P p y ´ y q “ v P p y q ă .As v P p a q “ e p P | p q v p p a q , we obtain that divides e p P | p q v p p a q . As p v p p a q , q “ , it follows that must divide e p P | p q . In particular, e p P | p q ě . As the other direction of the inequality holds by basicnumber theory, it follows that e p P | p q “ , and that p is fully ramified in L . We now prove that a place p of F q p x q is unramified in L whenever v p p a q ě . For the minimal polynomial f p X q “ X ´ X ´ a ,we have f p X q “ X ´ “ p X ´ q “ p X ´ qp X ` q . Let P be a place of L which lies above the place p of K . By [7, Theorem 3.5.10(a)], if P does notdivide f p y q “ p y ´ qp y ` q , then ď d p P | p q ď v P p f p y qq “ . Hence P would not appear in the COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 31 different, and p would be unramified in P . The same is also true for any root of f p X q ; these roots areprecisely y , σ p y q , and σ p y q . Furthermore, as v p p a q ě and each of y , σ p y q , and σ p y q is a root of f p X q , it follows that v P p y q , v P p σ p y qq and v P p σ p y qq are all nonnegative. Suppose then for the sake ofcontradiction that P divides p y ´ qp y ` q . Then it is necessary that P divides p y ´ q or p y ` q ,as P is prime. The place P must also divide p σ p y q ´ q or p σ p y q ` q , and p σ p y q ´ q or p σ p y q ` q (Ibid.). Let a , a , a P t´ , u such that P |p y ` a q , P |p σ p y q ` a q , and P |p σ p y q ` a q . Thus we have p y ` a q ` p σ p y q ` a q ` p σ p y q ` a q ” P . By construction, we have that the trace of f p X q is equal to y ` σ p y q ` σ p y q “ . We therefore obtain a ` a ` a “ p y ` a q ` p σ p y q ` a q ` p σ p y q ` a q ” P . As a , a , a P t´ . u and p ‰ , this is a contradiction. Thus P does not divide p y ´ qp y ` q and P does not appear in the different. As the constant field is finite, all places are separable, in particular p , and so by [8, Proposition 5.6.9], it follows that p is unramified in L .We now suppose that v p p a q ă and that | v p p a q . In particular we have that p ‰ p x , . We write a “ αγ β with α, β, γ P F p r x s , β is cube-free and p α, γ β q “ . It follows that p appears in γ , and that v p p α q “ v p p β q “ . The transformation y “ z { γ then yields ˆ z γ ˙ ´ ˆ z γ ˙ ´ αγ β “ , or z ´ γ z ´ αβ “ . Substituting u { β “ z gives us ˆ u β ˙ ´ γ ˆ u β ˙ ´ αβ “ , or u ´ γ β u ´ β α “ . The element u is integral over F q r x s . In particular, the F q r x s -ideal I u generated by t , u , u u iscontained in the integral closure O L of F q r x s in L . The determinant of the transformation matrix froman integral basis of O L to I u , which is equal to a constant multiple of ind p u q , is contained in F q r x s ,from which it then follows that ind p u q P F q r x s . We also have by definition thatdisc p u q “ ind p u q disc x p L q . Thus disc x p L q| disc p u q in F q r x s . By definition, the irreducible polynomial of u over K is equal to S p X q “ X ´ AX ` B where A “ γ β and B “ ´ β α . Thusdisc p S q : “ disc p u q “ A ´ B “ p γ β q ´ p´ β α q . As v p p α q “ v p p β q “ and v p p γ q ą , it follows by the strict triangle inequality that v p p disc p u qq “ .As disc x p L q| disc p u q in F q r x s , it follows that v p p disc x p L qq “ . As the constant field is finite, all placesare separable, in particular p , and so by [8, Proposition 5.6.9], it follows that p is unramified in L . (cid:3) By Lemma 5.9 and Theorem 5.10, we immediately obtain the following Corollary.
Corollary 5.12.
Suppose q ” ´ . Let L { F q p x q be a cubic extension and y a primitive elementwith f p y q “ y ´ y ´ a “ . Then the place p at infinity for x is unramified in L . Remark 5.13.
By Theorem 5.3, we know that only places corresponding to polynomials of evendegree can be ramified in cubic extensions with a primitive element y whose minimal equation is ofthe form y ´ y ´ a “ . Theorem 5.14 (Riemann-Hurwitz) . Suppose q ” ´ . Let L { F q p x q be a Galois cubic geometricextension and y a primitive element with f p y q “ y ´ y ´ a “ . Then the genus g L of L is givenaccording to the formula g L “ ´ ` ÿ v p p a qă p v p p a q , q“ deg p p q , where deg p p q denotes the degree of a place p of F q p x q .Proof. As r L : F q p x qs “ and L { F q p x q is Galois, it follows that all ramification indices are either equalto 1 or 3. Thus for a place p of F q p x q which ramifies in L , we have for P | p that, with p “ char p F q q , p e p P | p q , p q “ p , p q “ , whence by [8, Theorem 5.6.3], the differential exponent α p P | p q satisfies α p P | p q “ e p P | p q ´ “ . Furthermore, by Theorem 5.10, the places p of F q p x q which ramify in L are precisely those for which v p p a q ă and p v p p a q , q “ . We let d L denote the degree function on the divisors of L ; we thereforealso let d L p P q denote the degree of a place P of L . As L { F q p x q is of prime degree, it follows that d L p P q “ deg p p q for all ramified places p of F q p x q in L and P | p . We thus obtain by the Riemann-Hurwitz formula (see for example [Ibid., Theorem 9.4.2]) g L “ ` r L : F q p x qsp g F q p x q ´ q ` d L ´ D L { F q p x q ¯ “ ´ ` ÿ P | p v p p a qă p v p p a q , q“ d L p P q“ ´ ` ÿ v p p a qă p v p p a q , q“ deg p p q . Hence the result. (cid:3)
We now examine the valuation of a generator y of a cubic extension whose minimal equation is ofthe form y ´ y ´ a “ . Theorem 5.15.
Suppose that q ” ´ mod , and that L { F q p x q is cubic and Galois. For any place p of K , denote by P a place above p in L . We denote by p the place at infinity in F q p x q for x . Let y of L { F q p x q be a primitive element with minimal polynomial f p X q “ X ´ X ´ a where a P F q p x q . Let a “ α { γ β be the factorisation of a in F q p x q , where α, β, γ P F q r x s , p α, γ β q “ , and β is cube-free. Let P “ σ p P q and P “ σ p P q , where σ is a generator of Gal p L { F q p x qq . Then:(1) If p | β , then v P p y q “ v p p a q “ ´ v p p γ β q .(2) If p | γ and p ∤ β , then v P p y q “ ´ v p p γ q .(3) If p | α , then the places P , P , P are distinct, and exactly one element of t v P p y q , v P p y q , v P p y qu is equal to v p p a q , and the other valuations in this set are equal to .(4) For the place p :(a) If v p p a q “ , then v P p y q “ v P p y q “ v P p y q “ .(b) If v p p a q ą then the places t P , P , P u are distinct, exactly one of v P p y q , v P p y q , v P p y q is equal to v p p a q , and the two other valuations in this set are equal to .(5) For any other place p of K , v P p y q “ . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 33
Proof.
Suppose that p | β . By Theorem 5.11, we know that p is (fully) ramified in L . By the identity y ´ y “ a , it follows as in the proof of Theorem 5.11 that, for the unique place P of L above p , v P p y q ă . Thus v P p y q “ min t v P p y q , v P p y qu “ v P p y ´ y q “ v P p a q “ v p p a q , whence v P p y q “ v p p a q .Suppose that p | γ and p ∤ β . By Theorem 5.11, we know that p is unramified in L . Let P be a placeof L above p . Analogously to the previous case, we have that as v p p a q ă , so must v P p y q “ min t v P p y q , v P p y qu “ v P p y ´ y q “ v P p a q “ v p p a q ă , and thus that v P p y q “ v p p a q “ ´ v p p γ q “ ´ v p p γ q . Hence v P p y q “ ´ v p p γ q .Suppose that p | α . From the generating polynomial X ´ X ´ a for y and its conjugates σ p y q and σ p y q , we obtain by the non-Archimedean property that if v P p w q ą for one of w P t y , σ p y q , σ p y qu ,then v P p w q “ v p p a q .Furthermore, v p p a q ě implies that v P p y q ě , v P p σ p y qq ě , and v P p σ p y qq ě . By definition of theminimal polynomial for y , we have X ´ X ´ a “ p X ´ y qp X ´ σ p y qqp X ´ σ p y qq , and thus that ´ y ¨ p´ σ p y qq ¨ p´ σ p y qq “ ´ a , whence y σ p y q σ p y q “ a . Therefore, v p p a q “ v P p a q “ v P p y σ p y q σ p y qq “ v P p y q ` v P p σ p y qq ` v P p σ p y qq . As v P p w q P t , v p p a qu , therefore, we obtain that one, and only one, w P t y , σ p y q , σ p y qu has v P p w q ‰ ,for this w that v P p w q “ v p p a q , and that the valuations of the other two conjugates at P are equal to0. Note that v P p σ p w qq “ v σ p σ ´ p P qq p σ p w qq “ v σ ´ p P q p w q “ v P p w q and v P p σ p w qq “ v σ p σ ´ p P qq p σ p w qq “ v σ ´ p P q p w q “ v P p w q . Thus, as one of t v P p w q , v P p w qu is distinct from v P p w q , it follows that at least two of the places t P , P , P u are distinct. By [8, Corollary 5.2.23], as L { F q p x q is Galois, we have e f r “ r L : F q p x qs “ ,where e “ e p P | p q is the ramification index of P | p , f “ f p P | p q the inertia degree of P | p , and r thenumber of places of L above p . We have shown that r ą . As r | r L : F q p x qs “ , it follows that r “ ,and that P , P , and P are distinct. The result follows.For the place p “ p , we have by Lemma 5.10, that v p p a q ě . If v p p a q “ , then automatically v P p y q “ v P p y q “ v P p y q “ “ v p p a q . If v p p a q ą , then the valuation of y is positive at one, and only one, place P of L above p ;the proof of this is just the same as that for a place p dividing α , and for the place P , we obtain v P p y q “ v p p a q . (cid:3) Finally, we study the splitting behaviour of the unramified places for cubic extensions with a prim-itive element y whose minimal equation is of the form y ´ y ´ a “ . (Note that as r L : F q p x qs “ ,splitting is trivial for all (fully) ramified places.) Theorem 5.16.
Let q ” ´ , and let L { F q p x q be a Galois cubic geometric extension withgenerating equation X ´ X ´ a “ , where a P F q p x q . Let p be a place of F q p x q which is unramifiedin L and k p p q the residue field of F q p x q at p .(1) If v p p a q ą , then p is totally split in L . (2) If v p p a q ă , then v p p a q “ ´ m , the place p is finite, and | k p p q| ” . The place p splitscompletely in L if, and only if, the reduction f m p a mod p is a cube in k p p q where f p P F q r x s isthe irreducible polynomial associated with p . Otherwise, p is inert in L .(3) If v p p a q “ , then(a) If p ą , then p is inert in L if, and only if,(i) | k p p q| ” and p a ` δ q is not a cube in k p p q , where δ “ a ´ , or(ii) | k p p q| “ q “ and a “ ˘ .Furthermore, with a “ PQ with P , Q P F q r x s , p P , Q q “ , Let A , B P F q r x s relatively prime begiven as in Theorem 5.1 such that P “ p A ´ ´ B q and Q “ A ` ´ B . If v p p a q “ and | k p p q| ” , then p is inert if, and only if, A ` ?´ ´ BA ´ ?´ ´ B is not a cubemod p .(b) If p “ , then with | k p p q| “ n and tr : k p p q Ñ F the trace map tr p α q “ α ` α ` α ` ¨ ¨ ¨ ` α n ´ , the place p is inert in L if, and only if, tr p { a q “ tr p q and(i) | k p p q| ” and the roots of T ` aT ` are not cubes in F n “ k p p q , or(ii) | k p p q| ” and the roots of T ` aT ` are not cubes in F n “ k p p qp ?´ q .We note that the roots of T ` aT ` lie in F n , respectively, F n , depending on whether tr p { a q “ or tr p { a q “ .Proof. Throughout what follows in this proof, we will use P to denote a place of L above p .1. This is immediate from Theorem 5.15 (3) and (4).2. If v p p a q ă , then by Theorem 5.11 and Corollary 5.12, | v p p a q and the place p must be finite.Moreover, p is of even degree by Theorem 5.3, and as a consequence, | k p p q| ” mod . Let f p denote the irreducible polynomial corresponding to the place p , and let m P Z be such that v p p a q “ ´ m . Then, z “ f m p y is a root of the polynomial X ´ f m p X ´ f m p a , and v p p f m p a q “ . Inparticular, we have X ´ f m p X ´ f m p a ” X ´ f m p a mod p . Thus, p is inert in L if, and only if, X ´ f m p a is irreducible over k p p q , which occurs if, and only if, f m p a mod p P k p p q is not a cube in k p p q , and otherwise, as L { F q p x q is Galois of prime degree , p is completely split in L .3. If v p p a q “ , we let a P k p p q be the reduction of a modulo p .(a) We first suppose that p ą .i. Suppose that | k p p q| ” . Thus, ´ is a square in k p p q and k p p qp ?´ q “ k p p q . Thediscriminant of X ´ X ´ a is equal to ∆ “ ´ p a ´ q , and it is a square as L { F q p x q isGalois. As ´ is a square, the same is true for a ´ . Thus, there is δ P F q p x q such that a ´ “ δ . By [2, Theorem 3], the reduced polynomial X ´ X ´ a is irreducible over k p p q if, and only if, (1) its discriminant is a square in k p p q and (2) the element p a ` δ q is not acube in k p p q . As p q is always true as ∆ is a square, it follows that p is inert if, and only if, p a ` δ q is not a cube in k p p q . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 35
Moreover, by Theorem 5.1, writing a “ PQ with P , Q P F q r x s , p P , Q q “ , we know that thereexist coprime A , B P F q r x s such that P “ p A ´ ´ B q and Q “ A ` ´ B , whence ∆ “ A B p A ` ´ B q . By definition, we also have δ “ a ´ ” ´
13 16 A B p A ` ´ B q mod p , and hence δ ” ?´ ´ ˆ ABA ` ´ B ˙ mod p . Therefore, p a ` δ q ” ˜ p A ´ ´ B q A ` ´ B ` ?´ ˆ ABA ` ´ B ˙¸ mod p ” A ` ?´ ´ BA ´ ?´ ´ B mod p . ii. If | k p p q| ” , then again by [2, Theorem 3], the irreducible cubic polynomials in k p p qr X s are given by(4) X ´ X ν ´ p| k p p q|´ qp| k p p q|` q ´ ν ´ ν | k p p q| “ , where ν P k p p qp ?´ q is not a cube in k p p qp ?´ q . Thus, if X ´ X ´ a is an irreduciblepolynomial in k p p qr X s , then ν ´ p| k p p q|´ qp| k p p q|` q “ for a non-cube ν P k p p qp ?´ q . We also have p| k p p q| ´ qp| k p p q| ` q “ | k p p q| ´ | k p p q| ´ ă | k p p q| ´ . As ν P k p p qp ?´ q , it must also be true that ν is a p| k p p q| ´ q st root of unity. Thus p| k p p q| ´ qp| k p p q| ` q ˇˇˇ p| k p p q| ´ q “ p| k p p q| ´ qp| k p p q| ` q , so that p| k p p q| ´ q divides p| k p p q| ´ q , i.e., p| k p p q| ´ q ˇˇˇ p| k p p q| ´ q . As p| k p p q| ´ , | k p p q| ´ q “ , we obtain p| k p p q| ´ q| . As | k p p q| ě p ą , it follows that | k p p q| “ p “ . Hence “ ν ´ p| k p p q|´ qp| k p p q|` q “ ν ´ , so that by (4), a “ ν ` ν for the non-cube th root of unity ν . Let ζ P k p p qp ?´ q “ F be a primitive th root of unity. As ν is a non-cube in F , it follows that ν “ ζ i for some i “ , , , , and thus that ν is either a primitive rd or th root of unity. In the case that ν is a primitive rd root of unity, we have a “ ν ` ν “ ν ` ν “ ´ , whereas when ν is a primitive th root of unity, as ν “ ν ´ , we have ν ` ν “ ν ` ν p ν ´ q “ ν ´ ν ` ν “ ν p ν ´ q ´ p ν ´ q ` ν “ ν ´ ν ` “ . It follows that p is inert in L if, and only if, | k p p q| “ q “ and a “ ˘ .(b) If v p p a q “ and p “ , then by [9, Theorem 1], the polynomial X ´ X ´ a “ X ` X ` a P k p p qr X s is irreducible if over k p p q , and only if, tr p´ { a q “ tr p q and the roots of T ´ aT ` p´ q “ T ` aT ` P k p p qr T s are not cubes in k p p qp ?´ q . The result then follows by noting that ?´ P F n “ k p p q if, andonly if, | k p p q| ” , and otherwise that k p p qp ?´ q “ F n . (cid:3) Integral basis.
The next result gives an explicit integral basis for a Galois extension with gen-erating equation y ´ y ´ a “ . We treat the cases p ‰ and p “ separately within this Theorem,as discriminants exhibit different properties in each case. Theorem 5.17.
Let q “ ´ . Let L { F q p x q be a Galois cubic geometric extension with generator y which satisfies the equation y ´ y ´ a “ , where a P F q p x q . As before, we let a “ α {p γ β q where p α, βγ q “ and β is cube-free. Furthermore, let β “ β β , where β and β are squarefree, let O L bethe integral closure of F q r x s in L , and let ω “ γβ β y .(1) Suppose that p ‰ . Let A and B be as in Theorem 5.1. Then θ, κ P F q r x s may be chosen sothat θ ” ´ α p γ q ´ β ´ mod p AB q and θ ” γβ β mod β ,κ ” ´ p γβ β q mod p AB q β ,δ P F q r x s may be chosen freely, and the set I “ ! , ω ` δ, p AB q ´ β ´ p ω ` θω ` κ q ) forms a basis of O L over F q r x s .(2) Suppose that p “ . Let A and B be as in Theorem 5.1. An integral basis of the form B “t , ω ` S , p ω ` T ω ` R q{ I u exists for some S , T , R P F q r x s , where T “ γβ β ` A β H and H P F p r x s is chosen such that A ` B “ AG ` β γ β H , for some G P F q r x s . Remark 5.18.
Such a choice of H as in Theorem 5.17(2) always exists, as p B , β γβ q “ from p A , B q “ . Proof.
The element z “ γβ y satisfies the equation z ´ γ β z ´ αβ “ , and z is integral over F q r x s . Furthermore, for each finite place p of F q p x q , let f p P F q r x s be thepolynomial associated with p , and let β “ ź p | β v p p β q“ f p and β “ ź p | β v p p β q“ f p . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 37
Hence β “ β β , and the generating equation for z may therefore be written as z ´ γ β β z ´ αβ β “ . Division of this equation by β and replacing z with ω “ β ´ z “ γβ β y yields the equation ω ´ γ β β ω ´ αβ β “ . By definition, the discriminant of ω is equal to ∆ p ω q “ γ β β ´ α β β “ β β p γ β β ´ α q “ β β p γ β ´ α q . Let D L { F q p x q be the different of L { F q p x q [8, Section 5.6]. As the residue field extensions of a (fully)ramified prime p of F q p x q in K are trivial, it follows from the definition of D L { F q p x q and the fact thatthe place p is unramified in K that B L { F q p x q “ N L { F q p x q p D L { F q p x q q “ N L { F q p x q ¨˝ź P | β P ˛‚ “ ź P | p | β p f p P | p q “ ź p | β p “ pp β β q q F q r x s , where we let P denote a place of K above p .(1) Suppose that p ‰ . By Theorem 5.1, there exist coprime A , B P F q r x s such that α “ p A ´ ´ B q and γ β β “ γ β “ A ` ´ B . It follows that γ β ´ α “ p γ β q ´ α “ ` A ` ´ B ˘ ´ ` “ A ´ ´ B ‰˘ “ p AB q . Thus, ∆ p ω q “ β β p AB q . We wish to show that I “ ! , ω, p AB q ´ β ´ p ω ` θω ` κ q ) forms an integral basis of O K over F q r x s , where θ and κ are polynomials in F q r x s which arechosen (the former by the Chinese Remainder Theorem) so that θ ” ´ α p γ q ´ β ´ mod p AB q and θ ” γβ β mod β , and κ ” ´ p γβ β q mod p AB q β . We first prove that p AB q ´ β ´ p ω ` θω ` κ q is integral over F q r x s , i.e., ω ` θω ` κ ” AB β . We have p ω ` θω ` κ q “ ω ` θω ` p κ ` θ q ω ` θκω ` κ “ p ω ` θ q ω ` p κ ` θ q ω ` θκω ` κ “ p ω ` θ qp γ β β ω ` αβ β q ` p κ ` θ q ω ` θκω ` κ “ p γ β β ` κ ` θ q ω ` p αβ β ` θγ β β ` θκ q ω ` θαβ β ` κ . By definition of θ , we have θ ” ´ α p γ q ´ β ´ mod p AB q , and hence that θ ” α p γ q ´ β ´ mod p AB q ” γ β β ´ mod p AB q ” γ β β β ´ mod p AB q ” γ β β mod p AB q . Therefore γ β β ` κ ` θ “ γ β β ` κ ` θ ” γ β β ´ p γβ β q ` p γβ β q mod p AB q ” p AB q . Also by definition of θ , we obtain that θ ” p γβ β q mod β ” β . Thus, by definition of κ , it follows that γ β β ` κ ` θ “ γ β β ` κ ` θ mod β ” γ β β ´ p γβ β q mod β ” ´ γ β β mod β ” β . As p AB , β q “ , we have therefore proven that γ β β ` κ ` θ ” p AB q β . We also find that αβ β ` θγ β β ` θκ “ θ p γ β β ` κ q ` αβ β ” ´ α p γ q ´ β ´ p γ β β ´ p γβ β q q ` αβ β mod p AB q ” p´ α p γ q ´ β ´ p γβ β q q ` αβ β mod p AB q ” p AB q , and also, αβ β ` θγ β β ` θκ “ θ p γ β β ` κ q ` αβ β ” γβ β p γ β β ´ p γβ β q q ` αβ β mod β ” γ β β ` αβ β mod β ” p γ β β ` α q β β mod β “ β . We have therefore shown that αβ β ` θγ β β ` θκ ” p AB q β . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 39
Finally, we have θαβ β ` κ ” ´ α p γ q ´ β ´ αβ β ` p´ p γβ β q q mod p AB q ” ´ α β p γ q ´ ` γ β β mod p AB q ” β p γ β ´ α p γ q ´ q mod p AB q ” β p γ q ´ p γ β ´ α q mod p AB q ” p AB q . and θαβ β ` κ ” γβ β αβ β ` p´ p γβ β q q mod β ” γαβ β ` γ β β mod β ” β p γαβ ` γ β β q mod β ” β . We have thus proven that with θ and κ chosen as mentioned, p ω ` θω ` κ q “ p γ β β ` κ ` θ q ω ` p αβ β ` θγ β β ` θκ q ω ` θαβ β ` κ ” p AB q β , and hence that ω ` θω ` κ ” AB β . It follows that the element p AB q ´ β ´ p ω ` θω ` κ q is integral over F q r x s . As the extension L { F q p x q is of degree 3 and ω generates L over F q p x q , it follows that the three integral elements , ω ` δ , and p AB q ´ β ´ p ω ` θω ` κ q are linearly independent over F q p x q , and hence that I is a basis of L { F q p x q . Finally, the discriminant of the basis I is equal to ∆ p I q “ ¨˚˝ det ¨˚˝ δ p AB q ´ β ´ κ p AB q ´ β ´ θ p AB q ´ β ´ ˛‹‚˛‹‚ ∆ p ω q“ ¨˝ det ¨˝ p AB q ´ β ´ ˛‚˛‚ ∆ p ω q“ p AB q ´ β ´ ˆ β β p AB q ˙ “ p β β q , and hence that p ∆ p I qq F q r x s “ pp β β q q F q r x s “ B L { F q p x q . By basic theory (see for example [5, p. 398]), it follows that I is an integral basis for O L over F q r x s .(2) Suppose that p “ . By Theorem 5.1, there exist coprime A , B P F q r x s such that α “ A and γ β β “ γ β “ A ` AB ` B . It therefore follows from p “ that ∆ p ω q “ β β p γ β ´ α q “ β β A “ p β A q B L { F q p x q . By [5, Lemma 3.1, Corollary 3.2], a basis of the form B “ t , y ` S , p y ` Ty ` R q{ I u for some S , T , R P F q r x s exists, if and only if, there exists T P F q r x s such that T ` p γβ β q ” β A and T ` p γβ β q T ` αβ β ” β A , And if so, the set I “ " , ω ` T , I p ω ` T ω ` T ` p γβ β q q * forms an integral basis of L { F q p x q . We therefore investigate when such a T exists. First, wenote that the condition T ` p γβ β q ” β A is equivalent to T ” γβ β mod A and T ” β . This is equivalent to the existence of a polynomial H P F q r x s such that T “ γβ β ` A β H . Let us now choose such a T . Then, by definition we clearly have T ` p γβ β q T ` αβ β ” β . Moreover, T is invertible mod β , since p A , γβ β q “ . Thus, the condition T ` p γβ β q T ` αβ β ” A is equivalent to T ` p γβ β q T ` αβ β T ” A . We have T ` p γβ β q T ` αβ β T ”p γβ β q ` p γβ β q T ` αβ β T mod A ” β β p γ β β ` γ β T ` α T qq mod A Thus, the condition T ` p γβ β q T ` αβ β ” A is equivalent to γ β β ` γ β T ` α T ” A . Since p γ β , q “ , this condition is in turn equivalent to γ β β ` γ β T ` αγ β T ” A and B ` A B ` γ β T ` A γ β T ” A . p˚q That is, taking this equivalence mod A , we find in particular that B ` γ β T ” A . Hence γ β T ” B mod A . We write γ β T “ B ` AG where G P F q r x s . From this and p˚q , we obtain A G ` A G “ A G p G ` A q ” A . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 41
This implies in particular that G ” A . Thus, G “ AG for some G P F q r x s , so that γ β T “ B ` A G Since T “ γβ β ` A β H , we have γ β T “ γ β β ` A β γ β H . We therefore obtain the reduction B ` γ β β “ A p AG ` β γ β H q A ` AB “ A p AG ` β γ β H q A ` B “ AG ` β γ β H . As p A , β γ β q “ , such polynomials G and H must exist. In then follows by a similar argumentto that for the discriminant ∆ p I q in the proof of part (1) that the desired integral basis I exists,and that T “ γβ β ` A β H where H P F q r x s and G P F q r x s are chosen to satisfy A ` B “ AG ` β γ β H . Hence the result. (cid:3)
Appendix
Here, we restate Theorem 5.9 and give its proof.
Theorem 5.9 . Suppose that q ” ´ . Let L i “ K p z i q{ K ( i “ , ) be two cyclic extensions ofdegree such that z i ´ z i “ a i P K . The following are equivalent:(1) L “ L ;(2) z “ φ z ` χ z ´ φ , where φ, χ P K satisfy the equation χ ` a φχ ` φ “ . Moreover, if K “ F q p x q , when the above conditions are satisfied, then:(a) If p ‰ , there are relatively prime polynomials C , D P F q r x s φ “ CDQ δ p D ` ´ C q and χ “ ´ P CD ˘ δ p C ´ ´ D q δ p D ` ´ C q , where δ P K is such that δ “ Q d L { F q p x q and a “ P { Q , P , Q P F q r x s with p P , Q q “ and a “ p´ ` a q φ ` a φ χ ` φχ ` a χ . (b) If p “ ,(i) if φ “ , then χ “ ˘ . If χ “ , then a “ a , and if χ “ ´ , then a “ ´ a .(ii) if φ ‰ , then χ {p a φ q is a solution of X ´ X ´ φ ´ a φ “ and a “ a p a φ ` φ χ ` χ q . Proof.
Suppose that L “ L . As t , z , z u is a basis of L “ L over K , it follows that there are φ, χ, ψ P K such that z “ φ z ` χ z ` ψ. By Theorem 5.8, we have that σ p z q “ ´ uz ` f z ` u and σ p z q “ uz ` p´ ´ f q z ´ u for a generator σ of Gal p L { K q and where u “ ` fa , u “ f ` f ` , and f is one root of the polynomial S p X q “ ˜ ´ a ¸ x ` ˜ ´ a ¸ x ` ˜ ´ a ¸ . Thus, we have σ p z q “ φσ p z q ` χσ p z q ` ψ “ φ p´ uz ` f z ` u q ` χ p´ uz ` f z ` u q ` ψ “ p φ u ` φ f ´ u φ ´ χ u q z ` p φ u a ´ f u φ ` f χ q z ` p u φ ´ u f a φ ` u χ ` ψ q and σ p z q “ φσ p z q ` χσ p z q ` ψ “ φ p uz ´ p f ` q z ´ u q ` χ p uz ´ p f ` q z ´ u q ` ψ “ p φ u ` φ p f ` q ´ u φ ` χ u q z ` p φ u a ´ p f ` q u φ ´ χ p ` f qq z ` p u φ ´ u p f ` q a φ ´ u χ ` ψ q . As z satisfies z ´ z “ a , we have Tr p z q “ z ` σ p z q ` σ p z q “ . Thus, as ua “ f ` , we obtain “ z ` σ p z q ` σ p z q“ p φ ` φ u ` φ f ´ u φ ´ χ u ` φ u ` φ p f ` q ´ u φ ` χ u q z ` p χ ` φ u a ´ f u φ ` f χ ` φ u a ´ p f ` q u φ ´ χ p ` f qq z ` p ψ ` u φ ´ u f a φ ` u χ ` ψ ` u φ ´ u p f ` q a φ ´ u χ ` ψ q“ p φ ´ φ u ` φ f ` φ f q z ` p φ u a ´ f u φ ´ u φ q z ` p ψ ` u φ ´ u f a φ ´ ua φ q“ φ p´ u ` f ` f ` q z ` u φ p ua ´ f ´ q z ` p ψ ` u φ p u ´ f a ´ a q“ φ p´ u ` f ` f ` q z ` u φ p ua ´ f ´ q z ` p r ψ ` u φ p ´ a qq . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 43
This yields to the system $&% φ p´ u ` f ` f ` q “ u φ p ua ´ f ´ q “ ψ ` u φ p ´ a q “ The first two simplify to zero identically, which leaves only ψ ` u φ p ´ a q “ . As u a “ p f ` q and u “ f ` f ` , we obtain u p ´ a q “ u ´ a u “ p f ` f ` q ´ p f ` q “ ´ “ . (5)Thus the last equation of the previous system simplifies to “ ψ ` φ “ p ψ ` φ q , whence ψ “ ´ φ p‹q Inspection of the linear term yields z σ p z q ` z σ p z q ` σ p z q σ p z q “ ´ z σ p z q ´ z ´ σ p z q “ ´ . To simplify the computation, we write σ p z q “ φ z ` χ z ` ψ , with $&% φ “ ´ φ u ` φ f ´ χ u χ “ φ u a ´ f u φ ` f χψ “ u φ ´ u f a φ ` u χ ` ψ. p ˝ q We thus obtain “ z σ p z q ` z ` σ p z q “ p φ z ` χ z ` ψ qp φ z ` χ z ` ψ q ` p φ z ` χ z ` ψ q ` p φ z ` χ z ` ψ q “ p φφ ` φψ ` φφ ` ψφ ` φ ` χ ` φψ ` φ ` χ ` φ ψ q z ` p φφ a ` φχ ` χφ ` ψ χ ` ψχ ` φ a ` φχ ` χψ ` φ a ` φ χ ` χ ψ q z ` p φχ a ` χφ a ` ψψ ` ψ ` φχ a ` ψ ` φ χ a q . As a consequence, we obtain the system $’’&’’% “ φφ ` φψ ` χχ ` ψφ ` φ ` χ ` φψ ` φ ` χ ` φ ψ (i) “ φφ a ` φχ ` χφ ` ψ χ ` ψχ ` φ a ` φχ ` χψ ` φ a ` φ χ ` χ ψ (ii) “ φχ a ` χφ a ` ψψ ` ψ ` φχ a ` ψ ` φ χ a ´ (iii)We now work with Eq. (i). By definition, Eq. (i) simplifies to “ φφ ` φψ ` χχ ` ψφ ` φ ` χ ` φψ ` φ ` χ ` φ ψ “ φ p´ φ u ` φ f ´ χ u q ` ψ p´ φ u ` φ f ´ χ u q ` χ p φ u a ´ f u φ ` f χ q` φ p u φ ´ u f a φ ` u χ ` ψ q ` φ ` χ ` φψ ` p´ φ u ` φ f ´ χ u q ` p φ u a ´ f u φ ` f χ q ` p´ φ u ` φ f ´ χ u qp u φ ´ u f a φ ` u χ ` ψ q“ φ p u ` f ´ u f a ´ u ` ` f ` u a ` u f ´ u f a q` χ p f ` ´ u ` f q ´ u χψ ` φψ p ` f ´ u q ` φχ u p´ ` ua ´ f ´ u ´ f ` u f a q . Using that f ` f ` ´ u “ and ua “ f ` , the previous equation becomes “ φ p ` f ` f ` f ´ f p f ` q ´ p ` f ` f q ´ f p f ` q` p ` f ` f qp f ` q ` p ` f ` f q f ` ` f q ´ u χψ ´ f φψ ` φχ u p´ ` f ` ´ f ´ p f ` f ` q ´ f ` f p f ` qq“ ´ f φ ´ χψ u ´ f φψ ` φχ u p´ f ´ f ´ ´ f ` f ` f q“ ´ f φ ´ χψ u ´ f φψ ´ φχ u . Finally, as ψ “ ´ φ , the equation p i q´ f φ ´ χψ u ´ f φψ ´ φχ u “ ´ f φ ` φχ u ` f φ ´ φχ u “ is always satisfied.For Eq. (ii), we find “ φφ a ` φχ ` χφ ` ψ χ ` ψχ ` φ a ` φχ ` χψ ` φ a ` φ χ ` χ ψ “ φ p´ φ u ` φ f ´ χ u q a ` φ p φ u a ´ f u φ ` f χ q ` χ p´ φ u ` φ f ´ χ u q` p u φ ´ u f a φ ` u χ ` ψ q χ ` ψ p φ u a ´ f u φ ` f χ q ` φ a ` φχ ` χψ ` p´ φ u ` φ f ´ χ u q a ` p´ φ u ` φ f ´ χ u qp φ u a ´ f u φ ` f χ q` p φ u a ´ f u φ ` f χ qp u φ ´ u f a φ ` u χ ` ψ q“ φ p u a ` f a ´ f u ` a ` u a ´ u f a ´ f u ` f u a ` f a ´ u f q` χ p´ u ` u a ´ f u q ` rq p ` f q ` rp p´ f u ` u a q` φχ p´ ua ` f ` u ` f ` ´ u f a ` f u ´ f ua ` f q“ φ p u a ` f a ´ f u ` a ` u a ´ u f a ´ f u ` f u a ` f a ´ u f q` χ p´ u p f ` q ` u a q ` χψ p f ` q ` φψ p u a ´ f u q` φχ p´ ua ` f ` u ` f ` ´ u f a ` f u ´ f ua ` f q As f ` f ` ´ u “ and ua “ f ` , this simplifies to “ φ r u p f ` q ` f a ´ f u ` a ` u a ´ u f a ´ f u ` f u a ` f a ´ u f s` χψ p f ` q ` φψ r u p ua ´ f qs ` φχ r´p f ` q ` f ` f ` f ` ` f ` ´ f p f ` q ` f p f ` f ` q ´ f p f ` q ` f s“ φ r u a ` f a ´ f u ` a ` u a ´ u f a ´ f u ` f u a ` f a ´ u f s` ψχ p f ` q ` u φψ ` φχ p ` f q . “ φ p u a ` f a ´ f u ` a ` u a ´ u f a ´ f u ` f u a ` f a ´ u f ´ u q where the last equality holds as ψ “ ´ φ . By f “ ´ f ´ ` u and ua “ f ` , we therefore obtain “ u a ` f a ´ f u ` a ` u a ´ u f a ´ f u ` f u a ` f a ´ u f ´ u “ u a ` p´ f ´ ` u q a ´ f u ` a ` u a ´ u f a ´ f u ` p´ f ´ ` u q u a ` p´ f ´ ` u q a ´ u f p´ f ´ ` u q ´ u “ f a ´ u a ` f u ` u a ´ u f a ´ f u ´ u a f ` f a ` a ` u f ´ u COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 45 “ f a ´ u a ` f u ` u a ´ u f a ´ f u ´ u a f ` p´ f ´ ` u q a ` a ` u p´ f ´ ` u q ´ u “ ´ u a ´ f u ` u a ´ u f a ´ f u ´ u a f ´ u ` u “ ´ u p f ` q ´ f u ` u p f ` q ´ u f p f ` q ´ f u ´ u f p f ` q ´ u ` u “ ´ f u ´ u ` u ´ f u ´ f u ` u “ ´ f u ´ u ` u p f ` f ` q f ´ f u ´ f u ` u “ ´ f u ´ u ´ f u ` u “ ´ u p f ` f ` ´ u q , which is always satisfied.For Eq. (iii), we have “ φχ a ` χφ a ` ψψ ` ψ ` φχ a ` ψ ` φ χ a ´ “ φ a p φ u a ´ f u φ ` f χ q ` χ a p´ φ u ` φ f ´ χ u q ` ψ p u φ ´ u f a φ ` u χ ` ψ q` ψ ` φχ a ` p u φ ´ u f a φ ` u χ ` ψ q ` p´ φ u ` φ f ´ χ u qp φ u a ´ f u φ ` f χ q a ´ “ φ p´ a f u ´ a f u ` a u ` a f u ´ a f u ` u ´ a u q` φψ p´ a f u ` u q ` φχ p a ` a f ` a f ` a f ´ a u ´ a f u ` u ´ a u q` χ p´ a u ´ a f u ` u q ` χψ u ` ψ ´ . As ψ “ ´ φ , this becomes “ φ p´ a f u ´ a f u ` a u ` a f u ´ a f u ` u ´ a u q` φχ p a ` a f ` a f ` a f ´ a u ´ a f u ` u ´ a u q´ φ p´ a f u ` u q ` χ p´ a u ´ a f u ` u q ´ χφ u ` φ ´ “ φ p´ a f u ´ a f u ` a u ` a f u ´ a f u ` u ´ a u ` a f u ´ u ` q` φχ p a ` a f ` a f ` a f ´ a u ´ a f u ` u ´ a u ´ u q` χ p´ a u ´ a f u ` u q ´ . “ φ p´ a f u ` a u ` a f u ´ a f u ` u ´ a u ` a f u ´ u ` q` φχ p a ´ a u ´ a f u ` u ´ a u ´ u q ` u p ´ a q χ ´ . By Eq. (5), this is equal to φ p´ a f u ` a u ` a f u ´ a f u ` u ´ a u ` a f u ´ u ` q` φχ p a ´ a u ´ a f u ` a f ` u ´ a u ´ u q ` χ ´ “ As f ` f ` ´ u “ and ua “ f ` , the coefficient of φχ in Eq. (iii) becomes “ a ´ a u ´ a f u ` a f ` u ´ a u ´ u “ a ´ a u ´ a f u ` a f p´ f ´ ` u q ` u ´ a u ´ u “ a ´ a u ´ a f u ´ f a ´ f a ` f u a ` u ´ a u ´ u “ a ´ a u ´ a f u ´ a p´ f ´ ` u q ´ f a ` u ´ a u ´ u “ a ´ a u ´ a f u ` u ´ a u ´ u “ a ´ a u ´ a f u ` u ´ a u p f ` q ´ u “ a ´ a u ´ a f u ` u ´ u “ a ´ u p f ` q ´ p f ` q f u ` u p f ` f ` q ´ u “ a . Also as f ` f ` ´ u “ and ua “ f ` , the coefficient of φ in Eq. (vi) is equal to ´ a f u ` a u ` a f u ´ a f u ` u ´ a u ` a f u ´ u ` “ ´ f p f ` q ` p f ` q ` f p f ` q ´ f p f ` qp f ` f ` q ` p f ` f ` q ´ p f ` qp f ` f ` q` f p f ` q ´ p f ` f ` q ` “ . Therefore, Eq. (ii) becomes χ ` a φχ ` φ ´ “ , or equivalently, χ ` a φχ ` φ “ p‹‹q For the norm term in the equation z ´ z ´ a “ satisfied by z , we have ´ a “ ´ z σ p z q σ p z q “ z σ p z qp z ` σ p z qq . Hence ´ a “ z σ p z qp z ` σ p z qq“ p φ z ` χ z ` ψ qp φ z ` χ z ` ψ qpp φ ` φ q z ` p χ ` χ q z ` p ψ ` ψ qq“ φ p φ ` φ q φ z ` p χ φ ` φ φχ ` φ χ φ ` φ χ q z ` p ψ φ ` χ φχ ` φ φψ ` φ χ ` φ ψ φ ` χ φ ` φ χ χ ` φ ψ q z ` p ψ φχ ` χ φψ ` χ χ ` φ χψ ` ψ χ φ ` χ χ ` ψ φ χ ` χ φ ψ q z ` p ψ φψ ` ψ χ ` χ χψ ` φ ψ ` ψ φ ` χ ψ χ ` φ ψ ψ ` χ ψ q z ` p ψ χψ ` χ ψ ` ψ χ ` ψ χ ψ q z ` ψ p ψ ` ψ q ψ. By construction, z “ z ` a , so that the previous equation simplifies to “ p φ φ ` a χ φ ` ψ φ ` χ φχ ` a φ φχ ` φ φψ ` ψ φψ ` φ χ ` ψ χ ` χ χψ ` φ ψ ` φ φ ` a φ χ φ ` χ φ ` φ ψ φ ` ψ φ ` a φ χ ` φ χ χ ` χ ψ χ ` χ ψ ` φ ψ ` φ ψ ψ q z ` p φ a φ ` a ψ φ ` χ φ ` ψ φχ ` φ φχ ` a χ φχ ` χ φψ ` a φ χ ` χ χ ` ψ χψ ` φ χψ ` χ ψ ` φ χ φ ` ψ χ φ ` φ a φ ` a φ ψ φ ` a φ φψ ` a χ φ ` ψ φ χ ` a φ χ χ ` χ χ ` φ χ ` ψ χ ` ψ χ ψ ` χ φ ψ ` a φ ψ q z ` a χ φ ` a φ φχ ` φ a φ ` a χ χ ` a φ χψ ` ψ ψ ` a φ χ φ ` a ψ χ φ ` φ a φ ` a φ χ ` a ψ φ χ ` a χ χ ` ψ ψ ` a χ φ ’¸ R COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 47 ` a ψ φχ ` a χ φψ. As t z , z , u forms a basis of L { K , we obtain Eq. (iv), “ φ φ ` a χ φ ` ψ φ ` χ φχ ` a φ φχ ` φ φψ ` ψ φψ ` φ χ ` χ χψ ` φ ψ ` φ φ ` a φ χ φ ` χ φ ` φ ψ φ ` ψ φ ` a φ χ ` ψ χ ` φ χ χ ` χ ψ χ ` χ ψ ` φ ψ ` φ ψ ψ, Eq. (v), “ φ a φ ` a ψ φ ` χ φ ` ψ φχ ` φ φχ ` a χ φχ ` a φ χ ` χ χ ` ψ χψ ` φ χψ ` χ ψ ` φ χ φ ` ψ χ φ ` a χ φ ` ψ φ χ ` a φ χ χ ` χ χ ` φ χ ` ψ χ ` ψ χ ψ ` a φ ψ ` φ a φ ` a φ φψ ` a φ ψ φ ` χ φ ψ ` χ φψ, and Eq. (vi), ´ a “ a χ φ ` a φ φχ ` φ a φ ` a χ χ ` a φ χψ ` ψ ψ ` φ a φ ` a φ χ ` a ψ φ χ ` a χ χ ` ψ ψ ` a χ φ ψ ` a φ χ φ ` a ψ φχ ` a χ φψ ` a ψ χ φ. By the definitions of φ , χ , and ψ p ˝ q , Eq. (iv) becomes “ φ φ ` a χ φ ` ψ φ ` χ φχ ` a φ φχ ` φ φψ ` ψ φψ ` φ χ ` ψ χ ` χ χψ ` φ ψ ` φ φ ` a φ χ φ ` χ φ ` φ ψ φ ` ψ φ ` a φ χ ` φ χ χ ` χ ψ χ ` χ ψ ` φ ψ ` φ ψ ψ “ a p φ u a ´ f u φ ` f χ q φ ` p u φ ´ u f a φ ` u χ ` ψ q φ ` p φ u a ´ f u φ ` f χ q φχ ` a p´ φ u ` φ f ´ χ u q φχ ` p´ φ u ` φ f ´ χ u q φ ` p´ φ u ` φ f ´ χ u q φψ ` p´ φ u ` φ f ´ χ u q ψ ` p u φ ´ u f a φ ` u χ ` ψ q φψ ` p´ φ u ` φ f ´ χ u q χ ` p u φ ´ u f a φ ` u χ ` ψ q χ ` p φ u a ´ f u φ ` f χ q χψ ` p´ φ u ` φ f ´ χ u q φ ` a p´ φ u ` φ f ´ χ u qp φ u a ´ f u φ ` f χ q φ ` p φ u a ´ f u φ ` f χ q φ ` p´ φ u ` φ f ´ χ u qp u φ ´ u f a φ ` u χ ` ψ q φ ` p u φ ´ u f a φ ` u χ ` ψ q φ ` a p´ φ u ` φ f ´ χ u q χ ` p´ φ u ` φ f ´ χ u qp φ u a ´ f u φ ` f χ q χ ` p´ φ u ` φ f ´ χ u q ψ ` p φ u a ´ f u φ ` f χ qp u φ ´ u f a φ ` u χ ` ψ q χ ` p´ φ u ` φ f ´ χ u qp u φ ´ u f a φ ` u χ ` ψ q ψ ` p φ u a ´ f u φ ` f χ q ψ “ p a u ´ a u f ` f u ` u ´ a f u ` u ´ a f u ` a f u ` f ` a u ` f q φ ` p´ u ´ a u f ´ f u ` a u ` a f u ´ u f ´ u f ´ f u ´ u ´ a u ` a u ` a f ` a f ` a u ` a f ` a f q φ χ ` p u ´ a f u ` f ´ u ` f u ´ a f u ` f ` a u ` q φ ψ ` p f ` u ` f u ´ a u f ´ a f u ` f ` f ´ a u q φχ ` u p´ u ` a u ` a f u ´ ´ f ´ f q φχψ ` p´ u ` f ` q φψ ` u p a u ´ f ´ q χ ` p´ u ` f ` f ` q χ ψ ´ u χψ . As ψ “ ´ φ , this further simplifies to “ p´ u ´ f ` ` a f u ` u ` f u ´ a f u ` f ´ a u ´ a u f ` a f u ` a u q φ ` p u ´ a u ´ u ` f u ´ u f ´ a u f ` a f u ´ u f ´ f u ´ a u ` a f ` a f ` a u ` a f ` a f q φ χ ` p f ` u ` f u ´ a u f ´ a f u ` f ´ f ´ a u ´ q φχ ` u p a u ´ f ´ q χ . As f ` f ` “ u and a u “ f ` , the coefficient of φ in the previous expression is equal to ´ u ´ f ` ` a f u ` u ` f u ´ a f u ` f ´ a u ´ a u f ` a f u ` a u “ ´ p f ` f ` q ´ f ` ` f p f ` q ` p f ` f ` q ` f p f ` f ` q ´ f p f ` q ` f ´ p f ` q p f ` f ` q´ f p f ` qp f ` f ` q ` f p f ` q ` p f ` q “ . As f “ ´ f ´ ` u , the coefficient of φ χ is equal to u ´ a u ´ u ` f u ´ u f ´ a u f ` a f u ´ u f ´ f u ´ a u ` a f ` a f ` a u ` a f ` a f “ u ´ a u ´ u ` f u ´ u p´ f ´ ` u q ` a p´ f ´ ` u q u ´ p p´ f ´ ` u qq f u ´ a u ` a p´ f ´ ` u q ` a f p´ f ´ ` u q` au ` a p´ f ´ ` u q ` a f ´ a u f ´ u f “ p´ a ` u q f ` p´ a u ` u ´ u ´ a ´ a u q f ` u ´ a u ´ u ` a u ´ a ´ a u “ p´ a ` u qp´ f ´ ` u q ` p´ a u ` u ´ u ´ a ´ a u q f ` u ´ a u ´ u ` a u ´ a ´ a u “ ´ u p´ ` a u ` u ` a u q f ´ u p a u ` ´ u ´ u a ` a u q“ ´ u p´ ` p f ` q ` p f ` f ` q ` p f ` qq f ´ u p p f ` q` ´ p f ` f ` q ´ p p f ` qqp f ` f ` q ` p f ` q q“ For the same reason, the coefficient in χ φ is equal to f ` u ` f u ´ a u f ´ a f u ` f ´ f ´ a u ´ “ f ` p f ` f ` q ` f p f ` f ` q ´ f p f ` q COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 49 ´ f p f ` q ` f ´ f ´ p f ` q ´ “ Finally, the coefficient in χ is also , whence the Eq. (iv) is always satisfied.For Eq. (v), substitution of φ , χ and ψ via p ˝ q yields “ φ a φ ` a ψ φ ` χ φ ` ψ φχ ` φ φχ ` a χ φχ ` χ φψ ` a φ φψ ` a φ χ ` χ χ ` ψ χψ ` φ χψ ` χ ψ ` φ χ φ ` ψ χ φ ` φ a φ ` a χ φ ` ψ φ χ ` a φ χ χ ` χ χ ` φ χ ` ψ χ ` ψ χ ψ ` χ φ ψ ` a φ ψ ` a φ ψ φ “ p a u ` a f u ` a f ` a u ´ f u ` a u ´ a u f ` a f ´ a f u ´ u f ´ f u ´ a f u q φ ` p f ` a f u ` f ´ a u f ´ a u ´ a f u ´ a u f ´ a f u ` f u ` f u ` u ` f ` f ` u ` a u ` a u ` u a f ´ u a q φ χ ` p a ` a u ` a f u ` a f ´ u f ´ f u ` a u ` a f ´ f u ´ a u f q φ ψ ` p a u ´ u ` a u f ´ u ´ u f ` a f ` a f ´ f u ´ a u ` a f q φχ ` p f ` u ´ a f u ` f ` ´ a u f ´ a u ` f u ` f q φχψ ` u p a u ´ f q φψ ` p u ´ a f u ` f ´ a u ` f q χ ` u p a u ´ f ´ q χ ψ ` p ` f q χψ As ψ “ ´ φ , this becomes “ p a u ` f u ´ a ´ a u ` a f u ` a f ´ u f ´ f u ´ a f ´ a u f ` a u ´ a f u ´ a f u q φ ` p´ f ` a f u ` f ´ a u f ` a u ´ a u f ´ a f ` f u ` f u ` u ` f ´ f ´ u ` a u ` a u ` u a f ´ u a q φ χ ` p a u ´ u ` a u f ` u ´ u f ` a f ` a f ´ f u ´ a u ` a f q φχ ` p u ´ a f u ` f ´ a u ` f q χ Via u “ f ` f ` and ua “ f ` , we find that the coefficient in φ is equal to a u ` f u ´ a ´ a u ` a f u ` a f ´ u f ´ f u ´ a f ´ a u f ` a u ´ a f u ´ a f u “ a u ` f u ´ a ´ a u ` a f u ` a f ´ u f ´ f u ´ a f ´ a u f p f ` q ` a u p f ` q ´ a f p f ` q ´ a f p f ` q“ ´ a u ´ f u ` a p f ` q u ´ f p´ ` f q u ´ a p f ` f ` qp f ` q“ ´ a u ´ f u ` a p f ` q u ´ f p´ ` f q u ´ a u p f ` q“ ´ u p f ` q ´ f u ` p f ` q u p f ` q ´ f p´ ` f q u ´ u p f ` qp f ` q “ p´ f ´ q u ` p f ` qp f ` f ` q u “ p´ f ´ q u ` p f ` q u “ . Via the same relations, the coefficient in φ χ is equal to ´ f ` a f u ` f ´ a u f ` a u ´ a u f ´ a f ` f u ` f u ` u ` f ´ f ´ u ` a u ` a u ` u a f ´ u a “ ´ f ` f p ` f q ` f ´ f p ` f q ` p ` f q ´ f p ` f q´ f p ` f qp f ` f ` q ` f p f ` f ` q ` f p f ` f ` q` p f ` f ` q ` f ´ f ´ p f ` f ` q ` p ` f q p f ` f ` q` p ` f q ` f p ` f q ´ p ` f qp f ` f ` q“ . Similarly, we find that the coefficient in φχ is equal to a u ´ u ` a u f ` u ´ u f ` a f ` a f ´ f u ´ a u ` a f “ a u ´ u ` a u f ` u ´ u f ´ f u ´ a u “ p f ` q u ´ u p f ` f ` q ` p f ` q u f ` u ´ u f ´ f u ´ u p f ` q “ and finally, that the coefficient in χ is equal to u ´ a f u ` f ´ a u ` f “ p f ` f ` q ´ f p f ` q ` f ´ p f ` q ` f “ . For Eq. (ix), we substitute φ , χ , and ψ via p ˝ q to obtain ´ a “ a χ φ ` a φ φχ ` φ a φ ` a χ χ ` a φ χψ ` ψ ψ ` a φ χ φ ` φ a φ ` a φ χ ` a ψ φ χ ` a χ χ ` ψ ψ ` a χ φ ψ ` a ψ χ φ ` a χ φψ ` a ψ φχ “ ´ a p´ a u ` f u ´ a u ` f u ´ a u f ` f u ´ a f ` f a u ´ a f q φ ` a p u f ´ a u ` f ´ a u f ` f ` a u ` f u ´ u ´ a f u ` u ` f ´ a f u ` f q φ χ ` u p u ´ a f u ` ua f ´ a u ´ a f ` a u ´ f a q φ ψ ` a u p´ u ` a u ` a f u ´ f ´ ´ f q φχ ` p f a ` a ´ a u ` a f ` u ´ a f u ´ a u ` a f q φχψ ´ u p´ u ` f a q φψ ` a p f ´ u ` f q χ ´ u p´ u ` a ` f a q χ ψ ` u χψ ` ψ As ψ “ ´ φ , we obtain ´ a “ p´ f a ´ a ` a u ´ a f ´ u ` a f u ` a u ` a f ´ a u COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 51 ´ ua f ` a u ` a u f ´ a u ´ a f u ´ a f u ` a f ` u q φ χ ` u p a u ` a ´ a u ´ a f ` a f u ´ u q PQ ` a p f ´ u ` f q χ ` p u ´ a f u ´ u ` a u f ` a u ´ a f u ´ a u ` f a ´ f a u ` a f ´ q φ Via f ` f ` “ u , a u “ f ` , and ` a ´ ˘ f ` ` a ´ ˘ f ` ` a ´ ˘ “ , the coefficient of (ix)in φ is equal to u ´ a f u ´ u ` a u f ` a u ´ a f u ´ a u ` f a ´ f a u ` a f ´ “ p f ` f ` q ´ f p f ` q ´ p f ` f ` q ` f p f ` qp f ` f ` q ` a f ` p f ` q p f ` f ` q ´ f p f ` q ´ p f ` q ` a f ´ f p f ` q ´ “ p a ´ q f ` p a ´ q f ´ f ` “ ´ f p a ´ q ´ f ` “ ´ f a ` p f ` qp´ ` f q “ ´ f a ` p f ` qp a u ´ q “ p´ f ` u f ` u q a ´ p f ` q ua ` f ` “ p´ f ` u f ` u q a ´ p a u ´ f q ua ` f ` “ p´ u ´ f ` u f q a ` f ua ` f ` “ p´ u ` f p f ` qq a ` f p f ` q ` f ` “ p´ u ` f ` f q a ` f ` f ` “ p´ u ´ ` u q a ` p´ ` u q ` “ ´ u a ´ a ´ ` u “ ´ p f ` q ´ a ´ ` p f ` f ` q“ ´ a ` , and the coefficient in φχ is equal to u p a u ` a ´ a u ´ a f ` a f u ´ u q“ a u p f ` q ` a u ´ a u p f ` f ` q ´ a u f ` a u f p f ` q ´ u “ ´ u ` a u p f ` q“ ´ p f ` f ` q ` p f ` q “ ´ . The coefficient in φ χ is equal to ´ f a ´ a ` a u ´ a f ´ u ` a f u ` a u ` a f ´ a u ´ ua f ` a u ` a u f ´ a u ´ a f u ´ a f u ` a f ` u “ a f ´ p ua ´ q a f ´ p ua ´ u ` q a f ´ p ua ´ u ` q a f ` a u ` a u ´ ua ` a u ´ a ´ a u ´ u ` u “ a p´ f ´ ` u q ´ p ua ´ q a f p´ f ´ ` u q ´ p ua ´ u ` q a p´ f ´ ` u q´ p ua ´ u ` q a f ` a u ` a u ´ ua ` a u ´ a ´ a u ´ u ` u “ a p ua ` q f ´ a p u a ´ ua ´ u ´ q f ` a u ´ a u ` a ´ u ` u ´ a u ` ua ` a u “ a p ua ` qp´ f ´ ` u q ´ a p u a ´ ua ´ u ´ q f ` a u ´ a u ` a ´ u ` u ´ a u ` ua ` a u “ a u ` p´ f ` q a u ` au ` p f ` q a u ` p f ´ q au ´ u ` u “ a u ` p´ ua ` q a u ` a u ` a ua u ` p ua ´ q a u ´ u ` u “ ´ a p a ´ q u ` p a ´ q u ` a p´ ` a q u ` u , which by (5) yields a u ´ u ` a p´ ` a q u ` u “ a p a ´ q u “ ´ a , and finally, the coefficient in χ is equal to a p f ´ u ` f q “ a .As a conclusion, φ , χ , and ψ must only satisfy the following equations: $&% ψ “ ´ φ p q “ χ ` a φχ ` φ p q a “ p´ ` a q φ ` a φ χ ` φχ ` a χ p q When K “ F q p x q , we have the following:(a) If p ‰ , then we write φ “ UV with and U , V P F q r x s relatively prime. The quadratic equation (2)in χ , χ ` a φχ ` φ ´ “ , has a solution in F q p x q if, and only if, the discriminant Γ of the polynomial h p X q “ X ` a φ X ` φ ´ is a square in F q p x q . By definition, Γ “ a φ ´ p φ ´ q “ p a ´ q φ ` . We write a “ P { Q where P and Q P F q r x s and p P , Q q “ . We also write U “ gcd p U , Q q U and Q V “ gcd p Q , U q V , with U and V P F q r x s .Also, Γ is a square if, and only if, there exists W in F q r x s such that p P ´ Q q U ` V “ W . Multiplication by ´ yields Q d L { F q p x q U “ ´ p W ´ V q “ ´ p W ´ V qp W ` V q , where d L { F q p x q “ ´ p a ´ q is the discriminant of the minimal polynomial of z T p X q “ X ´ X ´ a . As L { F q p x q is Galois, it follows by Lemma 2.1 that d L { F q p x q P F q p x q . Let δ P F q r x s be suchthat δ “ Q d L { F q p x q . Thus p δ U q “ ´ p W ´ V qp W ` V q . As U and V are relatively prime,it follows that U , V , and W are pairwise coprime. Thus W ´ V and W ` V too are coprime.Therefore, by unique factorisation in F q r x s , it follows that, up to a unit, W ´ V and W ` V aresquares in F q r x s . Hence, there exist C , D P F q r x s relatively prime such that W ´ V “ ξ C and W ` V “ ζ D , where ξ, ζ P F ˚ q are such that ξζ is equal to ´ ´ up to a square in F ˚ q . Thus W “ ξ C ` ζ D and V “ ζ D ´ ξ C , so that W “ ζ ´ p ζξ C ` ζ D q and V “ ζ ´ p ζ D ´ ζξ C q . COMPLETE CLASSIFICATION OF CUBIC FUNCTION FIELDS OVER ANY FINITE FIELD 53
This is equivalent to W “ ζ ´ p´ ´ C ` D q and V “ ζ ´ p D ` ´ C q , where C “ ´ C and D “ ζ D . Therefore, we have W ´ V “ ´ ´ ζ ´ C and W ` V “ ζ ´ D , and it follows that ´ p W ´ V qp W ` V q “ ζ ´ D C “ δ U . By construction, we have δ U “ ˘ ζ ´ D C . As different values of ζ result in the same rational function φ , up to multiplication of C by ´ ,we have without loss of generality that δ U “ C D and Q V “ D ` ´ C , where C , D P F q r x s .For the converse, suppose that there exist C , D P F q r x s with C and D relatively prime such that Q V “ D ` ´ C and δ U “ CD . Then we find that φ “ CDQ δ p D ` ´ C q and Γ “ p a ´ q φ ` “ C D Q p P ´ Q q ` δ p D ` ´ C q δ p D ` ´ C q “ δ p´ ¨ ´ C D ` p D ` ´ C q q δ p D ` ´ C q “ p D ´ ´ C q p D ` ´ C q , which is a square in K . Hence, Γ “ γ with γ “ p D ´ ´ C q D ` ´ C , and χ “ ´ a φ ˘ γ “ ´ a φ ˘ p D ´ ´ C q D ` ´ C . (b) If p “ , then once more Eq. (2) holds. If φ “ , then Eq. (2) yields χ “ ˘ . By Eq. (3), if χ “ , then a “ a , whereas if χ “ ´ , then a “ ´ a . If φ ‰ , then as a ‰ , division of Eq.(2) by p a φ q yields ˆ χ a φ ˙ ` ˆ χ a φ ˙ ` φ ´ p a φ q “ . Thus, χ {p a φ q is a solution of the Artin-Schreier equation X ´ X ´ p φ ´ q{p a φ q “ , as claimed. (cid:3) Remark 5.19. (1) By Eq. (3), for the solution χ “ ´ a φ ` γ , we obtain furthermore that a “ p´ ` a q φ ` a φ χ ` φχ ` a χ “ p´ ` a q φ ` a φ p´ a φ ` γ q{ ` φ pp´ a φ ` γ q{ q ` a pp´ a φ ` γ q{ q “ ´ { p a ´ q φ ` { a p a ´ q φ γ ´ { p a ´ q φγ ` { a γ “ ´ { p a ´ q φ ` { a p a ´ q φ γ ´ { p a ´ q φ pp a ´ q φ ` q` { a γ pp a ´ q φ ` q“ ´ { p a ´ q φ ` { a p a ´ q φ γ ´ { p a ´ q φ ` { a γ “ ´ { p γ ´ qp a ´ q φ ` { a γ p γ ´ q . For the solution χ “ ´ a φ ´ γ , we obtain the analogous a “ p´ ` a q φ ` a φ χ ` φχ ` a χ “ p´ ` a q φ ` a φ p´ a φ ´ γ q{ ` φ pp´ a φ ´ γ q{ q ` a pp´ a φ ´ γ q{ q “ ´ { p a ´ q φ ´ { a p a ´ q φ γ ´ { p a ´ q φγ ´ { a γ “ ´ { p a ´ q φ ´ { a p a ´ q φ γ ´ { p a ´ q φ pp a ´ q φ ` q´ { a γ pp a ´ q φ ` q“ ´ { p a ´ q φ ´ { a p a ´ q φ γ ´ { p a ´ q φ ´ { a γ “ ´ { p γ ´ qp a ´ q φ ´ { a γ p γ ´ q . (2) In the language of Theorem 5.8, the base change matrix from t , z , z u to t , z , z u is givenby ¨˝ ´ φ χ φ φχ a ` φ φ a ` φχ ´ φ ` χ ˛‚ , and the inverse of this matrix is equal to ¨˚˚˝ ´ φ a φ ` χφ ´ χ φ ´ χ a φ ` χφ ´ χ pa φ ` χφ ´ χ φ p φχ ` a p φ ` χ q a φ ` χφ ´ χ φ p a φ ` χ q a φ ` χφ ´ χ ´ χ a φ ` χφ ´ χ ˛‹‹‚ . Therefore, z may be expressed in terms of z as z “ a φ ` χφ ´ χ “ ´ φ ` p φ ´ χ q z ` φ z ‰ . References [1] K. Conrad. Galois groups of cubics and quartics in all characteristics.
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