On small discriminants of number fields of degree 8 and 9
aa r X i v : . [ m a t h . N T ] D ec ON SMALL DISCRIMINANTS OF NUMBER FIELDS OF DEGREE 8 AND9
FRANCESCO BATTISTONI
Abstract.
We classify all the number fields with signature (4,2), (6,1), (1,4) and (3,3) havingdiscriminant lower than a specific upper bound. This completes the search for minimum dis-criminants for fields of degree 8 and continues it in the degree 9 case. We recall the theoreticaltools and the algorithmic steps upon which our procedure is based, then we focus on the nov-elties due to a new implementation of this process on the computer algebra system PARI/GP;finally, we make some remarks about the final results, among which the existence of a numberfield with signature (3 ,
3) and small discriminant which was not previously known. Introduction
Consider the family of number fields K with fixed degree n and fixed signature ( r , r ).Classical results by Minkowski and Hermite, obtained at the end of the XIX-th century, implythe following properties for the discriminant d K of the fields: • There exists an explicit lower bound | d K | > C ( n, r ), where C ( n, r ) > n and signature ( r , r ) there exists a field F such that | d K | attains the minimum valuewhen K = F . • For every
C >
0, there exist only finitely many number fields with fixed degree andsignature and such that | d K | ≤ C .The study of number fields with respect to their discriminants is then characterized by a doublepurpose: to find the minimum values for the discriminants of fields with fixed signature, and tocompletely classify all the fields in this family up to a chosen discriminant bound (a goal whichencompasses the first one). Complete tables of number fields with bounded discriminant areuseful tools in Number Theory, because they provide explicit examples over which one can getsome heuristic or prove results which are known to be asymptotically true in the discriminant(see, as an example, the work by Astudillo, Diaz y Diaz and Friedman [1] on minimum regulatorswhich explicitly requires this kind of lists).Giving a complete classification of fields with fixed degree n and signature ( r , r ) is easy for n = 2, because any quadratic field has the form Q ( √ d ) with d ∈ Z squarefree, and this structurereturns d K equal to either 4 d or d , depending on the residue class of d modulo 4.For n = 3 the research is still not difficult, thanks to Davenport-Heilbronn’s correspondancebetween isomorphism classes of cubic number fields and equivalence classes of primitive binaryintegral cubic forms [9], which was the theoretical cornerstone for Belabas’ algorithm for theclassification of cubic fields with bounded discriminant [4]. Mathematics Subject Classification.
Primary 11R21, 11R29, 11Y40.
Key words and phrases.
Number fields, classification for small discriminant.
Whenever one considers fields of higher degree, the classification becomes harder to get. Thereare two main mathematical frameworks, developed during the 70’s and 80’s of the XX-th century,which allowed several researchers to get results for fields with low degree: • Geometry of Numbers and its applications to the rings of integers, which provided explicitestimates on the possible maximum values for the coefficients of the defining polynomialsof number fields K with bounded | d K | : this was investigated by Hunter and Pohst [28]for number fields over Q and by Martinet [19] for generic number field extensions; • Lower bounds for the discriminants derived from the explicit formulae of Dedekind Zetafunctions, a procedure which was pursued by Odlyzko [20], Poitou [30] and Serre [34]and which allowed Diaz y Diaz [10] to obtain lower bounds of | d K | for several degreesand signatures.The simultaneous use of the previous tools permitted to develop algorithmic procedures whichgave complete classifications of number fields up to certain discriminant bounds in the followingcases: • Number fields with degree 4 [6] and 5 [31]; • Number fields with degree 6 [5, 21, 22, 23, 24, 25, 26] and degree 7 [11, 12, 14, 27]. • Totally complex [13] and totally real [29] number fields of degree 8; • Totally real number fields of degree 9 [36].For what concerns further signatures in degree 8, no complete tables up to some bound wereknown and for several years no attempts of this kind were made. During his Ph.D. work,the author [3] was then able to give a complete classification of number fields with degree8, signature (2 ,
3) and | d K | ≤ Theorem 1.
There exist 41 number fields K with signature (4 , and with | d K | ≤ .The minimum value of | d K | is .There exist 8 number fields K with signature (6 , and with | d K | ≤ . The minimumvalue of | d K | is .There exist 67 number fields K with signature (1 , and with | d K | ≤ . The minimumvalue of | d K | is .There exist 116 number fields K with signature (3 , and with | d K | ≤ . The minimumvalue of | d K | is . The number fields and the complete tables are gathered in the website [2], together with thePARI/GP programs used for their classification. The programs were run on the cluster systemof Universit´e de Bordeaux and on the clusters INDACO and HORIZON of Universit`a degli Studidi Milano.Here is an overview of the paper.
N SMALL DISCRIMINANTS OF NUMBER FIELDS OF DEGREE 8 AND 9 3
Section 2 recalls the theoretical foundations of the algorithmic procedure, which are respec-tively Hunter-Pohst-Martinet’s Theorem and the local corrections for lower bounds of discrim-inants given by prime ideals. Section 3 presents the various steps in which the algorithm isdivided. Section 4 finally presents the main novelties of our new implementation and someremarks on the final results obtained.
Acknowledgements.
I would like to thank my Ph.D. advisor Giuseppe Molteni for every use-ful suggestion and for his supervision of my research in Universit`a degli Studi di Milano. I wouldlike to thank also Institut de Math´ematiques de Bordeaux, for hosting me and allowing me touse the IMB cluster, and the people I worked with and gave me many advices: Bill Allombert,Karim Belabas, Henri Cohen, Andreas Enge, Aurel Page, Guillame Ricotta, Damien Robert.Thanks also to Alessio Alessi and Francesco Fichera, who gave me permission for using IN-DACO and HORIZON clusters respectively, to Scherhazade Selmane, for lending me her tableswith local corrections, and to Gunter Malle and Ken Yamamura, for their remarks about thepreliminary version of this paper. 2.
Theoretical recalls
In our procedure we look for irreducible monic polynomials of degree 8 and 9 with integercoefficients which define the desired number fields: the first problem consists then in givingan upper bound to the number of these polynomials, and the bound should depend on thediscriminant and the signature.Given a number field K , an element α ∈ O K and a number k ∈ Z , the Newton sum oforder k of α is defined as the sum S k ( α ) := n X i =1 α ki where the α i ’s represent the conjugates of α with respects to the embeddings σ , . . . , σ n of K .One has S k ( α ) ∈ Z , and S ( α ) = Tr( α ); moreover, if f ( x ) := x n + a x n − + · · · + a n − x + a n is the defining polynomial of α , then one has the recursive relations(1) S k ( α ) = − ka k − k − X j =1 a j S k − j ( α ) for every 2 ≤ k ≤ n which link the coefficients of f ( x ) to the values of the Newton sums.Consider then the absolute Newton sum T ( α ) := n X i =1 | α i | . We have an estimate for T ( α ), depending on n and | d K | , provided by Hunter-Pohst-Martinet’sTheorem [19]. F. BATTISTONI
Theorem 2.
Let K be a number field of degree n with discriminant d K . Then there exists anelement α ∈ O K \ Z which satisfies the following conditions:A) ≤ Tr( α ) ≤ j n k ; B) T ( α ) ≤ Tr( α ) n + γ n − (cid:18) | d K | n (cid:19) / ( n − =: U where γ n − is Hermite’s constant of dimension n − . The element α is called an HPM-element for K : for such an algebraic integer, the previoustheorem allows us to compute an upper bound for its trace and its second Newton sum S .These data, together with the absolute value of the norm N := | N( α ) | , are enough for givingupper bounds to every Newton sum, thanks to Pohst’s result [28]. Theorem 3.
Given K , U and α as in Theorem 2, given N ∈ N such that N ≤ ( U /n ) n/ , thenfor every k ∈ Z \ { , } we have an inequality | S k ( α ) | ≤ U k where U k is a number depending on n, r and U . Our goal is then to test the polynomials generated by a choice of the coefficients which derivesfrom the values of the Newton sums S k (with 2 ≤ k ≤ n ) ranging in the intervals [ − U k , U k ] andsatisfying the recursive relations (1). In order to do so, we need to choose an upper bound for | d K | . Remark 1.
The condition N ≤ ( U /n ) n/ is set in order to respect the inequality betweengeometric and arithmetic means: in fact, N = n Y i =1 | α i | ≤ (cid:18) P ni =1 | α i | n (cid:19) n = (cid:18) U n (cid:19) n . We recall now an inequality, proved by Poitou [30], which gives a lower bound for the dis-criminants of number fields with fixed degree n and signature ( r , r ). Theorem 4.
Let K be a number field of degree n , signature ( r , r ) and discriminant d K . Let f ( x ) be the function f ( x ) := (cid:18) x (sin x − x cos x ) (cid:19) . Then, for every y > , one has (2) 1 n log | d K | ≥ γ + log 4 π − L ( y ) − π n √ y + 4 n X p ⊂O K ∞ X m =1 log N( p )1 + (N( p )) m f ( m √ y log N( p )) where γ is Euler’s constant, the sum runs over the non-zero prime ideals of O K , N( p ) is theabsolute norm of the prime p and L ( y ) := L ( y ) + 13 L (cid:16) y (cid:17) + 15 L (cid:16) y (cid:17) + · · · + r n h L ( y ) − L (cid:16) y (cid:17) + L (cid:16) y (cid:17) · · · i where N SMALL DISCRIMINANTS OF NUMBER FIELDS OF DEGREE 8 AND 9 5 L ( y ) := − y + 3310 y + 2 + (cid:18) y + 34 y (cid:19) (cid:18) log(1 + 4 y ) − √ y arctan(2 √ y ) (cid:19) . Assume that we are able to guarantee that a prime ideal p with a fixed norm is contained in O K : this assumption provides then an explicit contribution to the estimate (2), which is called local correction for the discriminant given by an ideal of norm N( p ). We denote by C ( r , r , N( p )) the local corrections for fields with signature ( r , r ) given by a prime of normN( p ).Selmane [33] computed the values of local corrections for several signatures and prime ideals:in the following tables we report the lower bounds for | d K | obtained with local corrections forfields of degree 8 and 9, in every signature, and for prime ideals p of norm N( p ) ≤ Table 1.
Local corrections C ( r , r , N( p )) for number fields of degree 8( r , r ) (0,4) (2 ,
3) (4 ,
2) (6 ,
1) (8,0)N( p ) = 2 3379343 11725962 42765027 163060410 646844001N( p ) = 3 2403757 8336752 30393063 115852707 459467465N( p ) = 4 1930702 6688609 24363884 92810084 367892401N( p ) = 5 1656110 5726300 20829049 79259702 313918560N( p ) = 7 1362891 4682934 16957023 64309249 254052210 Table 2.
Local corrections C ( r , r , N( p )) for number fields of degree 9( r , r ) (1,4) (3 ,
3) (5 ,
2) (7 ,
1) (9,0)N( p ) = 2 81295493 301476699 1165734091 4679379812 19422150186N( p ) = 3 57789556 214235371 828172359 3323651196 13792634200N( p ) = 4 46348899 171694276 663330644 2660853331 11037921283N( p ) = 5 39657561 146723910 566314434 2269968332 9410709985N( p ) = 7 32371189 119294181 459066389 1835807996 7596751280Local corrections provide the following arithmetic consequences: if K has signature ( r , r )and | d K | < C ( r , r , N( p )), then O K does not admit any prime ideal with norm less or equalthan N( p ).This fact reflects then on the defining polynomials of the field: assume that | d K | ≤ C ( r , r , N( p )).If p ( x ) is a defining polynomial of K and α ∈ K is a root of p ( x ), then we know that | p ( n ) | = N(( α − n ) O K ) for every n ∈ Z , and so p ( n ) must not be an exact multiple of ev-ery m ∈ { , . . . , N( p ) } , i.e. m divides n and n/m is not divided by m .3. The Algorithmic Procedure
We want to detect all the number fields K with degree n , signature ( r , r ) and | d K | ≤ C ( r , r , C ( r , r ,
5) is the local correction for the signature ( r , r ) given by a primeideal of norm 5. We accomplish thus by constructing all the polynomials of degree n havinginteger coefficients bounded by the values U m obtained from Theorems 2 and 3 setting C ( r , r , F. BATTISTONI as upper bound of | d K | . Thannks to this construction, it is clear that we are dealing with definingpolynomials of HPM-elements.The polynomials are generated ranging the values for the Newton sums S m in the intervals[ − U m , U m ]; from these values we create the coefficients of the polynomials with the help ofthe recursive relations (1) and of further conditions derived from the arithmetic nature of theproblem, like the fact that any evaluation of the polynomial cannot be an exact multiple of 2,3, 4 or 5. Remark 2.
As stated above, the procedure assumes that we are looking for defining polynomialsof HPM-elements. There is a problem, however: unless the number field K is primitive, i.e.without subfields which are not Q and K , nothing assures us that the defining polynomial of anHPM element α ∈ K has degree exactly equal to n . In fact, α could be contained in a propersubfield of K .So we can just say that this procedure gives a complete classification only for primitive fields,which for composite degrees is still a proper subset of the considered family (though beingactually a very large subset).Fortunately, a relative version of Hunter-Pohst-Martinet’s Theorem [19] allowed to get acomplete classification of non-primitive fields up to larger upper bounds for | d K | , and specificallyin the following cases: • [8] and [32] give a complete classification of non-primitive fields of degree 8 with signature(2 , ,
2) and (6 ,
1) and | d K | ≤ , • [15] gives a classification of non-primitive number fields of degree 9 with | d K | ≤ · ,4 · , · , · , , · for the signatures (1 , , (3 , , (5 , , (7 , , (9 ,
0) respectively.Thus in our procedure we can restrict ourselves just to primitive fields.For what concerns the algorithmic procedure, this is in fact the same we used in order to classifynumber fields with signature (2,3), so that we wont give all the details here, but we will justrefer to what is presented in [3], Section 4. In fact, we obtained Theorem 1 by following theinstructions of the previous algorithm from Step 0 to Step 4.There are nonetheless some differences which must be remarked: first of all, one should replacethe previous upper bound 5762300 with the local correction C ( r , r , n = 8, like the amount of nested loopsor the checks done in Step 3, can be easily generalized for an arbitrary degree n .Next, there are some additional tests that can be made already in Step 1: the polynomial p ( x ) is kept if and only if it is constructed by Newton sums satisfying the followings restraints:If a = 0 , then S ≥ ,S ≥ − U + 2 n a , | S | ≤ (cid:18) S + U S + 2( U − S ) ) (cid:19) / , (3) S ≥ − U − S ) . The first two inequalities are proved in Cohen’s book [7], Chapter 9. Inequality (3) is proved bymeans of Cauchy-Schwartz inequality. The fourth inequality is a trivial necessary condition forthe validity of the third one
N SMALL DISCRIMINANTS OF NUMBER FIELDS OF DEGREE 8 AND 9 7 .Finally, Step 5 of the previous version is now put into Step 3, so that a candidate polynomial p ( x ) for defining a desired number field should satisfy, together with the conditions described inStep 3, the following properties. • p ( x ) must be an irreducible polynomial. • The field generated by p ( x ) must not have prime ideals of norm less or equal than 5.This can be verified in an algorithmic way (as we explain in the next section). Moreover,the signature of p ( x ) must be equal to ( r , r ). • Given an integer m , define coredisc( m ) as the discriminant of the number field Q ( √ m ).Then we require | coredisc(disc( p ( x ))) | < C ( r , r , . Once we have followed the instructions from Step 0 to Step 4, comprehensive of the above mod-ifications, one just needs to replace the previous Step 5 with the following Step 5’.
Step 5’:
We repeat the previous steps for every value of a between 0 and n/ a n which satisfies | a n | ≤ ( U /n ) n/ and is not an exact multiple of 2, 3, 4 or 5. Weare left with a list of polynomials among which we select the ones generating a number field K with signature ( r , r ) with | d K | ≤ C ( r , r , Remarks on the implementation and the results
The theoretical ideas on which our procedure is based and the several steps composing thealgorithm are very similar to what has been introduced in [3], with only few differences in some ofthe conditions put during the tests (like the check on the size of the coredisc). The main noveltywhich allowed us to obtain complete tables for further signatures is the different implementation,written only in PARI/GP, which gave the following consequences and facts: • As previously mentioned, the polynomials created during the process are tested alsoverifying that the ideals generated by the corresponding number fields do not have normless or equal than 5. The implementation of this process has been achieved thanks tothe
ZpX-primedec() function, written by Karim Belabas on purpose: the function istheoretically based upon the work by Ford, Pauli and Roblot ([16], Section 6) which usethe Round 4 Algorithm in order to recover the factorization of a prime ideal from the p -adic factorization of a minimal polynomial of the field.For what concerns its efficiency, this function is an order of magnitude faster than thepartial factorization given by nfinit() and faster than the usual decomposition function idealprimedec() : moreover, it is even faster whenever one deals with polynomials whichgive elements with small valuations for their indexes in O K . • The final check on the polynomials, suggested by Bill Allombert, concerns the size ofcore(disc( p ( x ))): this test was added only some month after the signatures in degree 8were solved. However, it allows to exclude many polynomials, because several candidatepolynomials p ( x ) have in fact core discriminants with very big size, which would forcethe number field discriminant to be way over the desired upper bound. F. BATTISTONI
The number of polynomials surviving this last condition is very small, being at most oforder 10 , and for these one can directly compute the number field discriminant. • The times of computation vary considerably and range from few hours (for signatures(2,3) and (4,2)), few days (signatures (6,1) and (1,4)), up to some months (signature(3,3)). • The tables presenting all the detected number fields can be found as PARI/GP filesat the website [2], together with the programs written by the author, the collection ofpolynomials found as result of the iterations and the overview on computation times.Finally, we present some remarks concerning the results described in Theorem 1. • Every field in our lists is uniquely characterized by its signature and the value of itsdiscriminant, with exceptions given only by two fields with signature (3 ,
3) and samediscriminant equal to − x − x − x + 2 x + 5 x + 6 x + 8 x + 4 x + 1 and x − x − x − x + 21 x + 35 x +23 x + 7 x + 1. • Every field of degree 8 and every field with signature (1 ,
4) contained in our lists wasalready known: in fact, they are all gathered into the Kl¨uners-Malle database of numberfields [18], although many of them miss from the LMFDB database [37]. Our workallows to say that these are the only number fields with the corresponding signatureswith discriminant less than the chosen upper bound. • Concerning the fields of degree 9 and signature (3 , | d K | ≤ − − • Every field in the list has trivial class group, and every primitive field has Galois group G of the Galois closure equal to either S or S , depending on the degree. More in detail,we have:27 primitive fields out of 41 fields with signature (4,2) (65.8% ca.).4 primitive fields out of 8 fields with signature (6,1) (50%).63 primitive fields out of 67 fields with signature (1,4) (94% ca.).112 primitive fields out of 116 fields in signature (3,3) (96.5% ca.).Furthermore, Table 3 provides the minimum values of | d K | for a field K with signature( r , r ) and with Galois group G as reported. • Although the algorithm classifies only primitive fields, every non-primitive field with | d K | ≤ C ( n, r ,
5) was displayed as output. • The groups in [2] are presented according to the LMFDB notation: every group isdenoted by n T q , where n is the degree of the corresponding field and q is the label ofthe group as transitive subgroup of S n : the choice of the label is based upon Hulpke’salgorithm for the classification of transitive subgroups of S n [17]. If the group has an N SMALL DISCRIMINANTS OF NUMBER FIELDS OF DEGREE 8 AND 9 9 n ( r , r ) G minimum | d K | S S S S Table 3.
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