Splitting of quaternions and octonions over purely inseparable extensions in characteristic 2
aa r X i v : . [ m a t h . R A ] D ec SPLITTING OF QUATERNIONS AND OCTONIONS OVERPURELY INSEPARABLE EXTENSIONS IN CHARACTERISTIC 2
DETLEV W. HOFFMANN
Abstract.
We give examples of quaternion and octonion division algebras overa field F of characteristic 2 that split over a purely inseparable extension E of F of degree ≥ F inside E of lowerexponent, or, in the case of octonions, over any simple subextension of F inside E . Thus, we get a negative answer to a question posed by Bernhard M¨uhlherrand Richard Weiss. We study this question in terms of the isotropy behaviour ofthe associated norm forms. Introduction
Recall that a quaternion (resp. octonion) algebra A over a field F is a an as-sociative 4-dimensional (resp. nonassociative 8-dimensional) composition algebrawhose associated norm form n A is a nondegenerate quadratic form defined on the F -vector space A . This norm form has the property that A is a division algebraiff n A is anisotropic. Furthermore, n A determines A in the sense that if A ′ is an-other quaternion (resp. octonion) algebra, then A ∼ = A ′ as F -algebras iff n A ∼ = n A ′ (i.e., the norm forms are isometric). There is a vast amount of literature on suchalgebras. In this note, we consider only base fields of characteristic 2, a case that,compared to characteristic not 2, gives rise to additional subtleties that make thetheory in some sense richer but also more intricate and complicated. For example,when conisdering subfields of such algebras one has to distingush between separableand inseparable quadratic extensions. The starting point for this investigation wasa question in this context posed to me by Bernhard M¨uhlherr und Richard Weiss.They asked the following: Given an octonion division algebra O over a field F ofcharacteristic 2 that splits over some purely inseparable algebraic extension E/F , isit true that then E contains a subfield K that is a quadratic extension of F and thatembeds into O ? We will show that the answer is negative in general by constructingvarious counterexamples.Our approach will be to consider this question in the context of quadratic formsusing the fact that the division property of such algebras can be expressed in termsof the anisotropy of the associated norm forms that are given by 2-fold Pfister formsin the case of quaternion algebras and by 3-fold Pfister forms in the case of octonionalgebras. This then allows to broaden the context to study the isotropy behaviourof arbitrary n -fold Pfister forms over purely inseparable extension in characteristic2. Mathematics Subject Classification.
Primary: 11E04; Secondary 11E81 12F15 16H0516K20 17A35 17A75.
Key words and phrases. quadratic form; Pfister form; isotropy; quaternion algebra; octonionalgebra; norm form; purely inseparable extension.The author is supported in part by DFG grant HO 4784/2-1
Quadratic forms, quadrics, sumsof squares and Kato’s cohomology in positive characteristic . In the next section, we will collect those facts about the theory of bilinear andquadratic forms in characteristic 2 that we will need in our constructions. In §
3, wewill study the isotropy behaviour of bilinear Pfister forms under purely inseparableextension and construct various examples that we will built upon in §
4, where weinvestigate the isotropy behaviour of quadratic Pfister forms under purely insepara-ble extensions. In that section, we will construct our main examples. Namely, givenany integers m ≥ n ≥ n ≥
2) and ℓ with 1 ≤ ℓ ≤ max { , m − } (resp.2 ≤ ℓ ≤ m ), we will construct a field F with an anisotropic n -fold quadratic Pfisterform π and a purely inseparable extension M/F such that M has exponent ℓ over F and [ M : F ] = 2 m and such that π will become isotropic over M but π will stayanisotropic over any simple (resp. exponent ≤ ℓ −
1) extension of F contained in M . In §
5, we will rephrase these constructions in the original context of quaternionand octonion algebras.2.
Quadratic and bilinear forms in characteristic ∼ =, ⊥ to denote isometry and orthogonal sum, and ⊗ for the tensorproduct of two bilinear forms or of a bilinear with a quadratic form. Let q (resp. b )be a quadratic (resp. bilinear form) on an F -vector space V . The value set is definedto be D F ( q ) = { q ( x ) | x ∈ V } ∩ F ∗ (resp. D F ( b ) = { b ( x, x )) | x ∈ V } ∩ F ∗ ), and q (resp. b ) is said to be isotropic if there exists some x ∈ V \ { } with q ( x ) = 0 (resp. b ( x, x ) = 0). If ( q ′ , V ′ ) is another quadratic form, then we say that q represents q ′ if there is a injective F -linear map σ : V ′ → V such that q ( σ ( x )) = q ′ ( x ) for all x ∈ V ′ , and we write q ′ ≺ q . Similarly, one defines b ′ ≺ b for another bilinear form( b ′ , V ′ ).Every quadratic form q decomposes as q ∼ = [ a , b ] ⊥ . . . ⊥ [ a r , b r ] ⊥ h c , . . . , c s i , a i , b i , c i ∈ F, where [ a, b ] stands for the quadratic form ax + xy + by and h c , . . . , c s i correspondsto the restriction of q to its radical and stands for the form c x + . . . + c s x s . Inparticular, the values s, r in such a decomposition are uniquely determined and q isnondegenerate (i.e., q has trivial radical) iff s = 0.If r = 0, we say that the above q is totally singular. In this case, D F ( q ) ∪ { } = P si =1 F c i is a finitely generated subvector space of F (considered as vector spaceover F ). In fact, if q ∼ = h c , . . . , c s i and q ′ ∼ = h d , . . . , d t i are both totally singular,then q ∼ = q ′ iff s = t and D F ( q ) ∪{ } = D F ( q ′ ) ∪{ } iff s = t and span F ( c , . . . , c s ) =span F ( d , . . . , d r ). We refer the reader to [8] concerning further properties of totallysingular quadratic forms. H = [0 ,
0] is called a quadratic hyperbolic plane, and q is said to be hyperboliciff q is isometric to an orthogonal sum of hyperbolic planes. If q ′ is nondegenerate,then q ′ ≺ q iff there exists a quadratic form q ′′ with q ∼ = q ′ ⊥ q ′′ . Also, it is notdifficult to show that if q is a nondegenerate quadratic form (or a quadratic formwith anisotropic radical), then q is isotropic iff H ≺ q .A 2-dimensional isotropic bilinear form is called a metabolic plane. For a suitablebasis, the Gram matrix will be of shape ( a ) for some a ∈ F , and this metabolic PLITTING OF QUATERNIONS AND OCTONIONS 3 form will be denoted by M a , and M = H b will be called a bilinear hyperbolic plane.A bilinear form b is called metabolic (hyperbolic) iff it is isomteric to an orthogonalsum of metabolic (bilinear hyperbolic) planes. A bilinear form is diagonalizable iff itis not hyperbolic, in which case we use the notation h a , . . . , a n i b when { e , . . . , e n } is an orthogonal basis of ( b, V ) with b ( e i , e j ) = δ ij a i ∈ F .An n -fold bilinear Pfister form ( n ≥
1) is a bilinear form that is isometric to h , a i b ⊗ . . . ⊗ h , a n i b for suitable a i ∈ F ∗ , in which case we write hh a , . . . , a n ii b forshort. h i b is considered to be the unique 0-fold bilinear Pfister form. The set offorms isometric to n -fold bilinear Pfister forms over F will be denoted by BP n F .An n -fold quadratic Pfister form ( n ≥
1) is a (nondegenerate) quadratic formthat is isometric to β ⊗ [1 , a ] for some a ∈ F and some β ∈ BP n − F . If β ∼ = hh a , . . . , a n − ii b , this quadratic Pfister form will be written hh a , . . . , a n − , a ]] forshort, and P n F will denote the set of forms isometric to n -fold quadratic Pfisterforms over F .We also consider totally singular quadratic forms that are so-called quasi-Pfisterforms of type hh a , . . . , a n ii ∼ = h , a i ⊗ . . . ⊗ h , a n i , a i ∈ F (here, we allow a i = 0).The set of quadratic forms isometric to n -fold quasi-Pfister forms will be denotedby QP n F . Note that D F ( q ) ∪ { } = F ( a , . . . , a n ) is just the field extension of F inside F generated by the a i , and that two forms in QP n F are isometric iff theseassociated fields are the same.If b ∈ BP n F , then the quadratic form q b given by q b ( x ) = b ( x, x ) is a form in QP n F . It should be noted that in this way, nonisometric forms in BP n F may yetgive rise to isometric forms in QP n F .Pfister forms play a central role in the theory of bilinear/quadratic forms and theyare characterized by many nice properties. For example, a bilinear (resp. quadratic) n -fold Pfister form ( n ≥
1) is metabolic (resp. hyperbolic) iff it is isotropic. Inparticular, if ϕ is a bilinear (quadratic) n -fold Pfister form for some n ≥
1, and if ψ is any other bilinear (quadratic) form with ψ ≺ aϕ for some a ∈ F ∗ and if inaddition dim ψ > dim ϕ (in which case ψ is called a Pfister neighbor of ϕ ), then ψ is isotropic iff ϕ is isotropic iff ϕ is metabolic (hyperbolic). Also a quadratic (resp.bilinear, resp. quasi) Pfister form π has the property that π ∼ = aπ for some a ∈ F ∗ iff a ∈ D F ( π ), a property that is often referred to as roundness.We are interested in the isotropy behaviour under purely inseparable algebraicextensions (p.i. extensions for short). We will recall a few facts regarding theseextensions in the next section.3. Bilinear forms over purely inseparable field extensions
Recall that if F is a field of characteristic 2, then elements a , . . . , a n ∈ F are called2-independent if [ F ( a , . . . , a n ) : F ] = 2 n , and a subset S ⊂ F is 2-independent ifany finite subset of S is 2-independent. A 2-basis of F is a 2-independent subset S with F ( S ) = F (such a 2-basis always exists).Let E/F be a p.i. extension. This means that to each a ∈ E there exists someinteger n ≥ a n ∈ F . The smallest such n will be called the exponent of a ,exp ( a ) = n , and we define the exponent of E to be exp ( E/F ) = sup { exp ( a ) | a ∈ E } . A p.i. extension E/F will be of exponent 1 iff there exists a nonempty 2-independent subset S ⊂ F such that E = F ( √ s | s ∈ S ). A finite p.i. extension E/F is called modular iff it is isomorphic to a tensor product of simple extensions
DETLEV W. HOFFMANN over F iff there exists a 2-independent subset { s , . . . , s r } ( r ≥
0) and integers n , . . . , n r ≥ E = F (cid:0) n √ s , . . . , nr √ s r (cid:1) . It is an easy exercise to show that if [ E : F ] = 2 m with m ≤ E/F is modular.But there are already examples of nonmodular extensions with [ E : F ] = 8 (we willencounter such an extension in Example 3.3). Proposition 3.1.
Let α ∼ = hh a , . . . , a n ii b ∈ BP n F and put A = { a , . . . , a n } ⊂ F ∗ .Let β be a bilinear form over F . Let E/F be a modular finite p.i. extension as in Eq.(3.1) for some -independent subset S = { s , . . . , s r } 6 = ∅ . Let L = F ( √ s , . . . , √ s r ) and σ ∼ = hh s , . . . , s r ii b . (i) α is anisotropic iff A is -independent. (ii) The following statements are equivalent: (a) β E is isotropic; (b) β L is isotropic; (c) σ ⊗ β is isotropic. (iii) α E is anisotropic iff A ∪ S is -independent.Proof. (i) See [5, Lem. 8.1].(ii) See [6, Th. 5.2].(iii) follows from (ii)(c) applied to α instead of β , together with (i) applied to thebilinear Pfister form σ ⊗ α ∼ = hh s , . . . , s r , a , . . . , a n ii b . (cid:3) α ∼ = hh a ii b is clearly anisotropic iff a ∈ F \ F . In this case, if E/F is any fieldextension, then α E is isotropic iff a ∈ E iff F ( √ a ) ⊂ E , so α becomes isotropic overa quadratic extension of F contained in E . This will generally no longer be trueonce we consider anisotropic bilinear Pfister forms of fold n + 2 ≥ Example . Let F be a field (of characteristic 2 as always) with a 2-independentset { x, y, z, c , . . . , c n } . For example, we may take F ( x, y, z, c , . . . , c n ), the rationalfunction field in n + 3 variables over F = Z / Z . Let E = F ( √ z, √ xz + y ) and π = hh x, y, c , . . . , c n ii b ∈ BP n +2 F . Clearly, { z, xz + y, x, y, c , . . . , c n } is 2-dependent.By Proposition 3.1, π is anisotropic over F and isotropic over E . Of course, thelatter can also be checked directly by noting that π ∼ = h , x, y, xy i b ⊥ h . . . i b and1 · √ xz + y + x · √ z + y · = 0 . We show that for any quadratic subextension F ⊂ K ⊂ E , we have that π K isanisotropic. Such a quadratic subextension is of shape K = F ( a ) with a = u √ z + v √ xz + y + w p z x + yz for some u, v, w ∈ F , not all equal to 0.By Proposition 3.1, π K is isotropic iff hh a , x, y, c , . . . , c n ii b is isotropic over F . Weknow that hh z, x, y, c , . . . , c n ii b is anisotropic over F . Now the (an)isotropy of thesebilinear Pfister forms is equivalent to the (an)isotropy of the associated quasi-Pfisterforms. We show that for these quasi-Pfister forms, we will have hh a , x, y, c , . . . , c n ii ∼ = hh z, x, y, c , . . . , c n ii , over F , which in turn implies that π K is anisotropic. PLITTING OF QUATERNIONS AND OCTONIONS 5
It cleary suffices to show that hh a , x, y ii ∼ = hh z, x, y ii , which just means that D F ( hh a , x, y ii ) ∪ { } = F ( a , x, y ) = F ( z, x, y ) = D F ( hh z, x, y ii ) ∪ { } . It then suffices to show that z ∈ F ( a , x, y ) and a ∈ F ( z, x, y ). Now a = ( wz ) x + v y + u z + v xz + w yz ∈ F ( z, x, y ) . Conversely, note that we do not have u = v = w = 0, and since x, y are 2-independent, we thus have 0 = s = u + v x + w y ∈ F ( a , x, y ). We put r = a + ( wz ) x + v y = z ( u + v x + w y ) = zs ∈ F ( a , x, y )and get that z = rs − ∈ F ( a , x, y ). (cid:3) In the next example, we will exhibit an anisotropic n -fold bilinear Pfister form( n ≥
2) that will become isotropic over a nonmodular p.i. extension but whichwill not become isotropic over any proper subfield. Now p.i. extensions
E/F with[ E : F ] ≤ Example . Let F be as in Example 3.2. Let E = F ( √ z, p x √ z + y ). Clearly,exp ( E/F ) = 2 and an easy check shows that [ E : F ] = 8. Comparing degreeand exponent shows that E/F is not simple. If
E/F were modular, then it wouldcontain a biquadratic subextension, i.e. an extension of degree 4 and exponent 1.However, the only exponent 1 subextension is F ( √ z ). We show this for the reader’sconvenience. Let ζ = √ z , χ = p x √ z + y and let t ∈ E with t ∈ F . Then thereare a i , b i ∈ F , 0 ≤ i ≤
3, such that t = X i =0 a i ζ i + χ X i =0 b i ζ i , and we get t = a + a z +( b z ) x + b xz + b y + b yz +( a + a z + b x + b xz + b y + b yz ) √ z ∈ F, so we get a + a z + b x + b xz + b y + b yz = 0which, by 2-independence, implies a = a = b = b = b = b = 0 and thus, t = a + a √ z ∈ F ( √ z ).Now let π ∼ = hh x, y, c , . . . , c n ii b be as in Example 3.2. Note that by generaltheory, the 2-independence of { z, x, y, c , . . . , c n } over F implies the 2-independenceof {√ z, x, y, c , . . . , c n } over L = F ( √ z ). Hence π L is anisotropic. Since E = L ( p √ z, p x √ z + y ), we see that we are in the same situation as in Example 3.2but with F replaced by L and z replaced by √ z . This shows on the one hand that π E is isotropic, and on the other hand that if K is any quadratic extension of L contained in E , then π K is anisotropic.We have that π is isotropic over E but anisotropic over any extension of F properlycontained in E . Indeed, if K is any extension of F properly contained in E , then K = F , or [ K : F ] = 2 in which case K = L , or [ K : F ] = 4 in which case K will contain a quadratic extension of F which must be L and K/L will thus be aquadratic extension. By the above, π K is anisotropic in all these cases. (cid:3) DETLEV W. HOFFMANN Quadratic forms over purely inseparable field extensions
The isotropy behaviour of quadratic forms over exponent 1 extensions has beenstudied in [7], including a determination of the Witt kernel for such extensions, i.e.,the classifiction of quadratic forms that become hyperbolic over such extensions.Complete results for quartic extensions can be found in [10], and the determinationof Witt kernels for arbitrary purely inseparable extensions can be found in [11], [2].We will not need the full thrust of these results but instead we will focus primarilyon quadratic Pfister forms and some explicit examples that have not been exhibitedbefore in the literature in the way we require.Let us remark at this point that 1-fold quadratic Pfister forms are of little interestin our context. Indeed, if π ∼ = hh a ]] ∼ = [1 , a ] ∈ P F is anisotropic and E/F is anyfield extension, then π E is isotropic iff a ∈ ℘ ( E ) = { e + e | e ∈ E } iff the separablequadratic extension F ( ℘ − ( a )) is contained in E (where ℘ − ( a ) denotes a root ofthe separable irreducible polynomial X + X + a ∈ F [ X ]). In particular, π will notbecome isotropic over any purely inseparable extension.The isotropy behaviour of quadratic forms over quadratic p.i. extensions is quitewell understood. Lemma 4.1.
Let q be an anisotropic quadratic form over F and let K = F ( √ a ) , a ∈ F \ F , be a quadratic p.i. extension. Then q K is isotropic iff c h , a i ≺ q forsome c ∈ F ∗ . In this case, hh a ii b ⊗ q is isotropic.Proof. The equivalence is well known (see, e.g., the proof of [9, Lemma 5.4]). Fur-thermore, if c h , a i ≺ q then c hh , a ii ∼ = c h , , a, a i ≺ hh a ii b ⊗ q , and h , , a, a i isobviously isotropic. (cid:3) Proposition 4.2.
Let
E/F be an exponent p.i. extension and let π ∈ P F beanisotropic. If π E is isotropic (and hence hyperbolic), there exist a, c ∈ F ∗ such that √ a ∈ E and π ∼ = hh a, c ]] . In particular, π becomes isotropic over a quadratic p.i.extension of F contained in E .Proof. The proof can be extracted from the proof of [7, Th. 3.4], but in our specialsituation, the argument can be condensed considerably and we include it for thereader’s convenience. Note that by assumption, E ⊆ F . Let π ∼ = hh u, v ]] ∈ P F be anisotropic but hyperbolic over E . Then any Pfister neighbor of π over F willalso become isotropic over E . Now [1 , u ] ⊥ h v i ≺ π ∼ = [1 , u ] ⊥ v [1 , u ], so there exist x, y, z ∈ E , not all equal to 0, with0 = x + xy + uy + vz . Then at least two of the x, y, z must be nonzero. If, say, x = 0, we may divide by x und thus, we may assume without loss of generality that we have an equation1 + y + uy + vz = 0 , y, z ∈ F. But y , z ∈ F and thus also y ∈ F . But then, the anisotropy over F implies that z ∈ E \ F . With z = a ∈ F it follows that π becomes hyperbolic over F ( √ a ) fromwhich we can conclude that π ∼ = hh a, c ]] for some c ∈ F (see, e.g., [1, Cor. 2.8]). Thecase y = 0 is similar. (cid:3) We will see later on that this result does not generalize to n -fold quadratic Pfisterforms for n ≥
3. In the case of n -fold quadratic Pfister forms ( n ≥ PLITTING OF QUATERNIONS AND OCTONIONS 7 will also not generalize to p.i. extensions of higher exponent, not even to simplesuch extensions (in stark contrast to the case of bilinear Pfister forms!) as the nextexamples will show.In the construction of our examples, we will often use generic methods. In partic-ular, we will work over the field of Laurent series F (( t )). Now theories of quadraticforms over valued fields in charactersitic 2 can be found in the literature, but we willnot need a full fledged such theory but rather only define some concepts and use somefacts that will be needed in our constructions to keep the paper as self-contained aspossible.We call a ∈ F (( t )) a unit if a ∈ F [[ t ]] ∗ = F ∗ + tF [[ t ]]. For our purposes, we call aquadratic form q over F (( t )) quasi-unimodular if ϕ has a representation ϕ ∼ = [ a , b ] ⊥ . . . ⊥ [ a r , b r ] ⊥ h c , . . . , c s i , a i , b i , c j ∈ F [[ t ]] ∗ ∪ { } (we include 0 to make sure that the hyperbolic plane [0 ,
0] will be quasi-unimodular).Then a i ∈ α i + tF [[ t ]] with some uniquely determined α i ∈ F (and similarly, b i ∈ β i + tF [[ t ]], c j ∈ γ j + tF [[ t ]] with β i , γ j ∈ F ). Then we get a quadratic form over F defined by ϕ = [ α , β ] ⊥ . . . ⊥ [ α r , β r ] ⊥ h γ , . . . , γ s i that we call the residue form for this representation. The following result is fairlystraightforward. Lemma 4.3.
Let ϕ be a quasi-unimodular quadratic form over F and let ϕ be theresidue form of some quasi-unimodular representation of ϕ . If ϕ is anisotropic then ϕ is anisotropic. In this situation, if ϕ ′ is the residue form of any other quasi-unimodular representation of ϕ , then ϕ ∼ = ϕ ′ over F . Proposition 4.4.
Let ρ , σ , α , β be quasi-unimodular quadratic forms over F (( t )) .Assume that α , β are totally singular of the same dimension n having quasi-uni-modular representations α ∼ = h a , . . . , a n i , β ∼ = h b , . . . , b n i . Let ψ ∼ = ρ ⊥ α,τ ∼ = σ ⊥ β,ϕ ∼ = ρ ⊥ t − σ ⊥ [ a , t − b ] ⊥ . . . [ a n , t − b n ] . If ψ and τ are anisotropic over F then ϕ is anisotropic over F (( t )) .Proof. The proof uses a fairly standard constant coefficient argument which we willonly sketch. Suppose ϕ is isotropic. Let dim ρ = r , dim σ = s . By multiplying by asuitable power of t , one may assume that there are e i , f j , g k , h k ∈ F [[ t ]] (1 ≤ i ≤ r ,1 ≤ j ≤ s , 1 ≤ k ≤ n ), not all divisible by t , such that0 = ρ ( e , . . . , e r ) + t − σ ( f , . . . , f s ) + n X k =1 (cid:0) a k g k + g k h k + t − b k h k (cid:1) . By considering the constant terms of the involved power series and by computingthe coefficient of t resp. t − in the above equation, one readily gets that we find anequation ρ ( ǫ , . . . , ǫ r ) + α ( δ , . . . , δ n ) = 0 with ǫ i , δ k ∈ F , not all equal to zero,or σ ( γ , . . . , γ s ) + β ( µ , . . . , µ n ) = 0 with γ j , µ k ∈ F , not all equal to zero, DETLEV W. HOFFMANN showing that ψ or τ is isotropic over F . (cid:3) Corollary 4.5.
Let a, a , . . . , a n ∈ F [[ t ]] ∗ such that b ∼ = h a , . . . , a n i b is an anisotropicbilinear form over F (( t )) and with associated totally singular quasi-unimodular qua-dratic form q b ∼ = h a , . . . , a n i . If q b is anisotropic over F then b ⊗ [1 , at − ] isanisotropic over F (( t )) .Proof. Note that b is anisotropic iff the associated quadratic form q b is anisotropic.Also, b ⊗ [1 , at − ] ∼ = n ⊥ i =1 a i [1 , at − ] ∼ = n ⊥ i =1 [ a i , aa − i t − ] , and we can apply Proposition 4.4 to ψ ∼ = α and τ ∼ = β ∼ = a h a − , . . . , a − n i ∼ = a h a , . . . , a n i ∼ = α. (cid:3) In the next example, we extend Example 3.2 by showing that to each n ≥
0, thereare examples of fields over which there exist anisotropic ( n +3)-fold quadratic Pfisterforms that become isotropic over a biquadratic p.i. extension but stay anisotropicover any quadratic subextension. Example . Let F , E , π be as in Example 3.2, put L = F (( t )), M = L ( √ z, √ xz + y ) = F ( √ z, √ xz + y )(( t )) = E (( t ))and q = π ⊗ [1 , t − ] ∈ P n +3 L . Obviously, hh z, x, y, c , . . . , c n ii is quasi-unimodularwith anisotropic residue form (which essentially is the form with the same coeffi-cients), and by Corollary 4.5, it follows that hh z ii b ⊗ q is anisotropic and thus also q .But as in Example 3.2, π E and hence π M and q M are isotropic.Let K be any quadratic extension of L contained in M . As in Example 3.2, wemay write K = L ( a ) where a = u √ z + v √ xz + y + w p z x + yz with u, v, w ∈ L , not all equal to 0. Multiplying by a suitable power of t , we mayfurthermore assume that u = u + u ′ , v = v + v ′ , w = w + w ′ with u , v , w ∈ F ,not all equal to 0, and u ′ , v ′ , w ′ ∈ tF [[ t ]]. Let a = u √ z + v √ xz + y + w p z x + yz. Then a ∈ a + tF [[ t ]]. Using the 2-independence of x, y, z , and the fact that we donot have u = v = w = 0, we have a = 0 and thus a ∈ F [[ t ]] ∗ . As in Example3.2, we get (over F !) hh a , x, y, c , . . . , c n ii ∼ = hh z, x, y, c , . . . , c n ii which is anisotropic over F .By the above, hh a , x, y, c , . . . , c n ii is quasi-unimodular with anisotropic residueform hh a , x, y, c , . . . , c n ii . By Corollary 4.5, it follows that hh a , x, y, c , . . . , c n ii b ⊗ [1 , t − ] ∼ = hh a ii b ⊗ q is anisotropic over L . Therefore, with L ( a ) = K and by Lemma 4.1, q K is anisotropic. (cid:3) PLITTING OF QUATERNIONS AND OCTONIONS 9
Example . In the previous example, we have constructed fields L over which thereexist an anisotropic ( n + 3)-fold Pfister form π ( n ≥
0) over L and a p.i. extension M of L of degree 4 such that exp ( M/L ) = 1 and π becomes isotropic over M but staysanisotropic over any simple extension K/L contained in E (since in that example, M/L is p.i. biquadratic, any such simple subextensions will be p.i. quadratic).One can easily generalize this as follows:
For any integers n ≥ , m ≥ and ℓ with ≤ ℓ ≤ max { , m − } there exist a field L , an anisotropic ( n + 3) -fold Pfister form π over L and a p.i. extension M of L ofdegree m and exponent exp ( M/L ) = ℓ such that π becomes isotropic over M but π stays anisotropic over any simple extension K of L contained in M . Indeed, choose integers r ≥ ≤ m , . . . , m r ≤ ℓ such that m = 2 + m + . . . + m r and m = ℓ if m ≥ L = F (( t ))where this time F has a 2-independent set { z, x, y, c , . . . , c n , b , . . . , b r } . Again, we choose π ∼ = hh x, y, c, . . . , c n , t − ]] over L This time, we define F ′ = F (cid:0) m √ b , . . . , mr √ b r (cid:1) L ′ = F ′ (( t )) M = L (cid:0) √ z, √ xz + y, m √ b , . . . , mr √ b r (cid:1) Note that M = L ′ ( √ z, √ xz + y ) = F ′ ( √ z, √ xz + y )(( t )) . By 2-independence, one easily checks that the p.i. extension
M/L satisfies the re-quired properties regarding degree and exponent. Furthermore, by the general the-ory of 2-independence, (cid:8) z, x, y, c , . . . , c n , m p b , . . . , mr p b r (cid:9) is a 2-independent set over F ′ (and over L ′ ), thus, we can now proceed exactly asin Example 4.6 and conclude that π L ′ is anisotropic, π M is isotropic, and π K ′ isanisotropic over any simple extension K ′ of L ′ contained in M .Now let K = L ( w ) be any simple extension of L contained in M . Then obviously K ′ := KL ′ = L ′ ( w ) is a simple extension of L ′ contained in M , hence π K ′ isanisotropic and thus also π K since K ⊆ K ′ . (cid:3) Example . Let F , E , π be as in Example 3.3, put L = F (( t )), M = L ( √ z, p x √ z + y ) = F ( √ z, p x √ z + y )(( t )) = E (( t ))and q = π ⊗ [1 , t − ] ∈ P n +3 L . As in Example 4.8, M/L is nonmodular of degree 8, andthe only quadratic extension of L inside M is L ( √ z ) = F ( √ z )(( t )). As in Example4.6, the 2-independence of {√ z, x, y, c . . . , c n } over L ( √ z ) implies the anisotropyof q L ( √ z ) , and M/L ( √ z ) is a biquadratic extension. Combining the arguments inExample 4.6 with those in Example 3.3 shows that if K is any extension of L properlycontained in M then q K is anisotropic. We leave the details to the reader. (cid:3) We now want to construct examples of anisotropic ( n + 2)-fold Pfister forms overa field F that become isotropic over a simple p.i. extension E/F of exponent m ≥ ≤ m −
1. For this,we collect two lemmas that we will need in our construction.The first lemma is well known (and easy to prove). We assume as usual that wehave characteristic 2, but of course it holds more generally in any positive charac-teristic p . Lemma 4.9.
Let F be a field and m ≥ be an integer. Let x ∈ F \ F and E = F (cid:0) m √ x (cid:1) . Then [ E : F ] = 2 m and all the intermediate fields are given in thefollowing filtration of successive quadratic p.i. extensions: F ⊂ F (cid:0) √ x (cid:1) ⊂ F (cid:0) √ x (cid:1) ⊂ . . . ⊂ F (cid:0) m − √ x (cid:1) ⊂ F (cid:0) m √ x (cid:1) = E. The next lemma provides some relations for quadratic forms over fields of char-acteristic 2.
Lemma 4.10.
Let u, v ∈ F ∗ and m ≥ be an integer. (i) hh u, uv ]] ∼ = hh v, uv ]] . In particular, hh u, uv ]] will be isotropic over F ( √ v ) . (ii) [1 , u m ] ∼ = [1 , u ] .Proof. (i) We have hh u, uv ]] ∼ = [1 , uv ] ⊥ u [1 , uv ] ∼ = [1 , uv ] ⊥ [ u, v ] ∼ = [1 , uv ] ⊥ v [1 , uv ] ∼ = hh v, uv ]] . Therefore, h , v i ≺ hh u, uv ]] will be isotropic over F ( √ v ).(ii) This follows since u m ≡ u mod ℘ ( F ) (essentially, this is a comparison of theso-called Arf-invariants of the two forms). (cid:3) Example . Let F be a field (of characteristic 2 as usual) with elements a , . . . , a n , x, y ∈ F ∗ ( n ≥ m ≥ ξ = m − √ x, E ′ = F ( ξ ) and E = E ′ ( p ξ ) = F (cid:0) m √ x (cid:1) . We define the Pfister form π ∼ = hh a , . . . , a n , y, y m − x ]] . Note that π will be isotropic over E since over E ′ , we have by Lemma 4.10(ii) that hh y, y m − x ]] E ′ ∼ = hh y, ( yξ ) m − ]] E ′ ∼ = hh y, yξ ]] E ′ which is isotropic over E = E ′ ( √ ξ ) by Lemma 4.10(i).If we now can find a field F with elements a i , x, y as above such that [ E : F ] = 2 m (or, equivalently, x ∈ F \ F ) and π E ′ anisotropic, then we have our example: π will be an anisotropic ( n + 2)-fold Pfister form over F that will become isotropicover a simple p.i. extension E with exp ( E/F ) = m , but, by Lemma 4.9, π will stayanisotropic over any extension of F of exponent ≤ m − E .For this, let F be a field with 2-independent elements a , . . . , a n , y ∈ F ∗ and put E = F (( t )) and F = F (( t m )) . Furthermore, put x = t − m . Clearly, E = F ( m √ x ). We put τ = t so that the above ξ will become ξ = t − = τ − . We thus get the filtration F = F (( t m )) ⊂ F (( t m − )) ⊂ . . . ⊂ F (( τ )) = E ′ ⊂ F (( t )) = E ′ ( p ξ ) = E of all intermediate fields between F and E (cf. Lemma 4.9). PLITTING OF QUATERNIONS AND OCTONIONS 11
Since a , . . . , a n , y ∈ F ∗ are 2-independent, the form hh a , . . . , a n , y ii b is anisotropicover F . Also, over E ′ = F (( τ )) and by Lemma 4.10(ii), we have (cid:2) , y m − x (cid:3) ∼ = (cid:2) , ( yτ − ) m − (cid:3) ∼ = (cid:2) , yτ − (cid:3) , we can apply Corollary 4.5 to conclude that π E ′ ∼ = hh a , . . . , a n , y ii b ⊗ (cid:2) , y m − x (cid:3) ∼ = hh a , . . . , a n , y ii b ⊗ (cid:2) , yτ − (cid:3) is anisotropic over E ′ (cid:3) Example . In the previous example, we have constructed a field F with ananisotropic ( n + 2)-fold Pfister form, n ≥
0, and a simple p.i. extension
E/F with[ E : F ] = 2 m and exp ( E/F ) = m ≥ π becomes isotropic over E but π will stay anisotropic over any extension K/F of exponent ≤ m − E .Again, this can be generalized as follows: Given any integers n ≥ , m ≥ ℓ ≥ , there exists a field F with an anisotropic ( n + 2) -fold Pfister form π over F and with a p.i. extension M/F with [ M : F ] =2 m and exp ( M/F ) = ℓ such that π M is isotropic, but π K is anisotropic for anyextension K of F contained in M with exp ( K/F ) ≤ ℓ − . Indeed, let r ≥ ≤ m , . . . , m r ≤ ℓ be integers with m = ℓ + m + . . . + m r (this is always possible under the assumptions). Let us proceed as in Example 4.11by choosing a field F but this time with a 2-independent set { y, a , . . . , a n , b , . . . , b r } , we put F = F (( t ℓ )) and x = t − ℓ and π ∼ = hh a , . . . , a n , y, y ℓ − x ]]. Let F ′ = F (cid:0) m √ b , . . . , mr √ b r (cid:1) F ′ = F ′ (( t ℓ )) M = F (cid:0) ℓ √ x, m √ b , . . . , mr √ b r (cid:1) Note that M = F ′ (cid:0) ℓ √ x (cid:1) . Using 2-independence, one readily checks that the p.i.extension M/F satisfies [ M : F ] = 2 m and exp ( M/F ) = ℓ .Since (cid:8) y, a , . . . , a n , m √ b , . . . , mr √ b r (cid:9) is a 2-independent set over F ′ (cf. Example 4.7), we conclude as in Example 4.11that π M will be isotropic, but π K ′ will be anisotropic over any extension K ′ of F ′ contained in M with exp ( K ′ /F ′ ) ≤ ℓ − m = ℓ and r = 0).Now let K be any extension of F contained in M with exp ( K/F ) ≤ ℓ −
1. Thenclearly exp ( KF ′ /F ′ ) ≤ ℓ −
1, so by the above, π stays anisotropic over KF ′ andthus over K . (cid:3) Applications to quaternions and octonions
We refer to [12] concerning basic properties of quaternions and octonions that wewill state in the sequel. Let a, b ∈ F ∗ , c ∈ F . Recall that a quaternion algebra Q = ( b, c ] F over such a field F of characteristic 2 is a 4-dimensional central simple(associative) algebra generated by two elements i, j subject to the relations i + i = c, j = b, ji = ( i + 1) j.Q carries an F -linear involution σ given by σ ( i ) = i + 1, σ ( j ) = j . An octonionalgebra O = ( a, b, c ] F over F is an 8-dimensional nonassociative composition algebrathat can be gotten from a quaternion algebra Q as above through the Cayley-Dicksonconstruction by defining O = Q ⊕ Qℓ where the product is given by( x ⊕ yℓ )( z ⊕ wℓ ) = ( xz + aσ ( w ) y ) ⊕ ( wx + yσ ( z )) ℓ, x, y, z, w ∈ Q.O carries an involution τ given by τ ( x ⊕ yℓ ) = σ ( x ) ⊕ yℓ. On Q (resp. O ) one has a norm given by N Q ( x ) = xσ ( x ) (resp. N O ( x ) = xτ ( x ))which defines a quadratic Pfister form on Q (resp. O ) given by η Q ∼ = hh b, c ]] (resp. η O ∼ = hh a, b, c ]]).It is well known that the isometry class of the norm form determines the isomor-phism class of the quaternion (resp. octonipn algebra). That means, if A i , i = 1 , η i = η A i , then A ∼ = A as algebras iff η ∼ = η as quadratic forms. Furthermore, A i is division iff η i is anisotropic.Using this correspondence between the division property of quaternion/octonionalgebras and the anisotropy of their associated norm forms and by invoking ourExamples 4.7 and 4.12, we now give our version of the negative answer to thequestion posed by M¨uhlherr and Weiss from the introduction. Corollary 5.1.
To any integers m ≥ and ℓ with ≤ ℓ ≤ max { , m − } (resp. ≤ ℓ ≤ m ), there exists a field F of characteristic together with a p.i. extension M/F of degree m and exponent ℓ , and with an octonion division algebra O (resp. aquaternion division algebra Q and an octonion division algebra O ) over F such that O M is split (resp. Q M and O M are split), but O (resp. Q and O ) will stay divisionover any simple (resp. exponent ≤ ℓ − ) extension K of F contained in M . References [1] Ahmad, H.;
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Structure of inseparable extensions , Ann. of Math. (2) (1968), 401–410. Fakult¨at f¨ur Mathematik, Technische Universit¨at Dortmund, D-44221 Dortmund,Germany
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