Square-Central and Artin-Schreier Elements in Division Algebras

0.08 MB·English

by  Demba Barry

Square-Central and Artin-Schreier Elements in Division
Algebras
5
DembaBarrya,AdamChapmanb
1
0
2 aDe´partementdeMathe´matiqueetInformatique,Universite´deBamako,CollinedeBadalabougou
BPE3206,Bamako,Mali
r
p bDepartmentofMathematics,MichiganStateUniversity,EastLansing,MI48824
A

3
1

Abstract
]
A
We study the behavior of square-central elements and Artin-Schreier elements in
R
division algebras of exponent 2 and degree a power of 2. We provide chain lem-
.
h mas for such elements in division algebras over 2-fields F of cohomological 2-
t
a dimension cd (F) ≤ 2, and deduce a common slot lemma for tensor products of
2
m
quaternion algebras oversuch fields. Wealso extend to characteristic 2atheorem
[ proven by Merkurjev for characteristic not 2 on the decomposition of any central

2 simplealgebraofexponent2anddegreeapowerof2overafieldFwithcd (F) ≤ 2
2
v
asatensorproductofquaternion algebras.
1
3
Keywords: Quaternion algebras, Quadratic forms,Commonslotlemma
8
2010MSC: primary16K20;secondary 11E04,11R52
3
0
.
1
0 1. Introduction
5
1 Aquaternion algebra overafield F isanalgebra oftheform
:
v
i [α,β) = F[x,y : x2+ x= α,y2 = β,yxy−1 = x+1]
X
r
a forsomeα ∈ F andβ∈ F× ifchar(F) = 2,and
(α,β) = F[x,y : x2 = α,y2 = β,yxy−1 = −x]
forsomeα,β∈ F× ifchar(F) , 2.
The common slot lemmafor quaternion algebras states that for every twoiso-
morphic quaternion algebras (α,β) and (α′,β′) there exists β′′ ∈ F× such that
Emailaddresses: [email protected](DembaBarry),[email protected]
(AdamChapman)
PreprintsubmittedtoArchivderMathematik April14,2015
(α,β) (cid:27) (α,β′′) (cid:27) (α′,β′′) (cid:27) (α′,β′) (see [Jac96, Lemma 5.6.45 & Corollary
5.6.48]).
Anoncentral element vinaquaternion F-algebra issquare-central ifv2 ∈ F×.
If char(F) = 2, a noncentral element v is Artin-Schreier if v2 +v ∈ F. For every
twoArtin-Schreier elements vandwinaquaternion algebra there existsasquare-
centralelementtsuchthatvt+tv = tw+wt = t. Ifchar(F) , 2,thesquare-central
elements are the “pure quaternions”. Forevery twosquare-central elements v and
w in a quaternion algebra there exists another square-central element t such that
vt = −tv and tw = −wt. The common slot lemma is an immediate result of this
fact, which we refer to as the chain lemma for square-central or Artin-Schreier
elements.
Weextend thenotions ofsquare-central and Artin-Schreier elements andtheir
chainlemmastoanyalgebraofdegreeapowerof2andexponent2. Thenotionof
commonslotlemmaextendsnaturallytoanytensorproductofquaternionalgebras.
Incharacteristicnot2,acommonslotlemmaandachainlemmaforsquare-central
elements in tensor products of two quaternion algebras were provided in [Siv12]
and [CV13], respectively. Equivalent results in characteristic 2 were provided in
[Cha13, Section 3.3]. In [Cha15] a common slot lemma was provided for tensor
products of quaternion algebras over fields F with cd (F) ≤ 2. However, a chain
2
lemmaforthesquare-centralelementswasnotprovided,andthelengthofthechain
wasnotcomputed.
In this paper we prove several facts about square-central and Artin-Schreier
elements,andinparticularweprovideachainlemmaforthemindivisionalgebras
of exponent 2 over 2-fields F with cd (F) ≤ 2. We then deduce the common slot
2
lemma for tensor products of quaternion algebras over such fields and bound the
lengthofthechainsfromaboveby3,whichisequaltothemaximallengthofsuch
chains for biquaternion algebras over arbitrary fields. We make use of [Kah90,
Theorem 3]onthedecomposition ofanycentralsimplealgebraofexponent 2and
degreeapowerof2overafieldFwithcd (F) ≤ 2asatensorproductofquaternion
2
algebras, which was proven in that paper for characteristic not 2 and we prove it
herealsoforcharacteristic 2.
2. Preliminaries
Quadratic forms play a major role in the study of central simple algebras of
exponent 2. By I F we denote the group of Witt equivalence classes of even di-
q
mensional nonsingular quadratic forms over the field F. Every such form is iso-
metric to [a ,b ] ⊥ ··· ⊥ [a ,b ] for some integer m and a ,b ,...,a ,b ∈ F
1 1 m m 1 1 m m
if char(F) = 2, and ha ,...,a i for some a ,...,a ∈ F× if char(F) , 2. The
1 2m 1 2m
2
expression [a,b] stands for the quadratic form au2 +uv+bv2, ha ,...,a i stands
1 n
forthediagonal forma u2+···+a u2,and⊥istheorthogonal sumofforms.
1 1 n n
Thediscriminant(alsoknownastheArfinvariantincharacteristic2)isdenoted
by δ and defined as follows: If char(F) = 2, δ maps I F to the additive group
q
F/{a2+a : a∈ F}by
δ([a ,b ] ⊥ ···⊥ [a ,b ]) = a b +···+a b .
1 1 m m 1 1 m m
Ifchar(F) , 2,δmaps I F tothemultiplicative group F×/(F×)2 by
q
δ(ha ,...,a i) = (−1)ma ...a
1 2m 1 2m
(see[EKM08,Section13]).
Thesubgroup of I F offormswithtrivial discriminant isdenoted by I2F. We
q q
followthetraditional abuseofnotation ofwriting f ∈ I2F whenweactuallymean
q
thattheWittequivalence classof f belongs toI2F.
q
TheCliffordalgebraofaquadratic form f isdefinedtobe
C(f)= F[x ,...,x :(u x +···+u x )2 =
1 2m 1 1 2m 2m
= f(u ,...,u ) ∀u ,...,u ∈ F].
1 2m 1 2m
For f ∈ I F withdim(f) = 2m,thisalgebraisacentralsimplealgebraofexponent
q
2 and degree 2m. The Clifford invariant c(f) of f is the Brauer class of C(f) in
Br (F). Restricted to I2F, the Clifford invariant is an epimorphism from I2F to
2 q q
Br (F)withkernelI3F(see[Mer81]forchar(F) , 2and[Sah72]forchar(F) = 2).
2 q
For f ∈ I2F one hasC(f) (cid:27) M (E(f)) with E(f) a central simple algebra over F
q 2
ofdegree2m−1 isomorphic toatensorproduct ofquaternion algebras.
Thecohomological 2-dimension of F, denoted by cd (F), isbydefinition ≤ 2
2
ifHn+1,n(L,Z/2Z) = 0foreveryfinitefieldextensionL/F andn ≥ 2(see[EKM08,
Section 101.B]). Thelatterholds ifandonly if I3L = 0(see[EKM08,Fact16.2]).
q
Ifcd (F) ≤ 2then I2F (cid:27) Br (F).
2 q 2
A field extension K/F is excellent if for every form f over F there exists a
quadratic form f′ over F such that f′ is isometric to the anisotropic part of f .
K K
In particular every quadratic field extension is excellent (see [EKM08, Example
29.2] and [MM95, Lemma 1]). A 2-field is a field with no nontrivial odd degree
extensions. (In [EKM08] such fields are called 2-special.) In [Mer91, Theorem
4] interesting examples of fields F with cd (F) = 2 were constructed. The odd
2
closure F′ ofsuch F alsohascd (F′) = 2(see[EKM08,Example101.17]).
2
Thefollowinglemmawillbeusedlateron:
Lemma 2.1. For any field extension K/F, if f ∈ I F and f ∈ I2K then there
q K q
exists f′ ∈ I2F suchthat f′ ≃ f .
q K K
3
Proof. Letδbearepresentative ofthediscriminant of f.
Assume char(F) = 2. On the one hand δ ∈ F. On the other hand, δ = u2 +u
for some u ∈ K because f ∈ I2K. Now, f ≃ [a ,b ] ⊥ ··· ⊥ [a ,b ] for some
K q 1 1 n n
a ,b ,...,a ,b ∈ F. Without loss of generality we can assume a , 0. Set
1 1 n n 1
f′ = [a ,b +a−1δ] ⊥ ··· ⊥ [a ,b ]. Becauseofthediscriminant, f′ ∈ I2F.
1 1 1 n n q
Weshallnowprovethat f′ ≃ f . Sinceallthesummandsarethesameexcept
K K
the first one, weshall prove that under the scalar extension, the firstsummands in
bothformsareisometric.
[a ,b +a−1δ] = a x2+ xy+(b +a−1δ)y2 ≃ a (x2+ xy+(a b +δ)y2) ≃
1 1 1 K 1 1 1 1 1 1
a ((x+uy)2+(x+uy)y+(a b +δ)y2) = a (x2+ xy+a b y2)≃ [a ,b ]
1 1 1 1 1 1 1 1 K
(Thefirstisometryisobtained byreplacing ywitha y.)
1
Assume char(F) , 2. Onthe one hand δ ∈ F×. Onthe other hand, δ = u2 for
some u ∈ K× because f ∈ I2K. Now, f ≃ ha ,...,a i for somea ,...,a ∈ F×.
K q 1 n 1 n
Set f′ = hδ−1a ,a ...,a i. Because of the discriminant, f′ ∈ I2F. It is obvious
1 2 n q
that f′ ≃ f . (cid:3)
K K
3. Square-centralandArtin-Schreierelements
LetAbeadivisionalgebraofexponent2anddegree2n overafieldF forsome
n ≥ 2. In this section we prove a couple of facts about square-central and Artin-
Schreier elements in A without restricting the cohomological dimension of F. If
char(F) , 2 and xis square-central then any element t inthe algebra decomposes
into t + t such that t = 1(t + xtx−1) commutes with x and t = 1(t − xtx−1)
0 1 0 2 1 2
anti-commutes with x. Ifchar(F) = 2and xisArtin-Schreierthenanyelementtin
thealgebradecomposes intot +t suchthatt = xt+tx+tcommuteswith xand
0 1 0
t = xt+txsatisfies xt +t x= t .
1 1 1 1
Lemma3.1. AssumeAisadivisionalgebraofdegree4andexponent2. Let xand
x′ betwocommutingelementsinAwithF[x] , F[x′]. Ifchar(F) = 2thenassume
xiseitherArtin-Schreierorsquare-central and x′ isArtin-Schreier. Ifchar(F) , 2
assumethat xand x′ aresquare-central. ThenAdecomposesasQ ⊗Q suchthat
1 2
x ∈ Q and x′ ∈ Q .
1 2
Proof. Assume char(F) = 2. The involution on F[x,x′] mapping x′ to x′ + 1
and x to x+ 1 if x is Artin-Schreier and to x if x is square-central extends to A.
Denote this involution by τ. Then τ restricts to an involution of the second kind
on the centralizer C(x′) (a quaternion algebra over F[x′]), and it commutes with
thecanonical involution γ onthisalgebra. Therefore, γ◦τisanautomorphism of
4
order2onC(x′),andthealgebraoffixedpointsisaquaternion algebra Q over F
1
containing xandcentralizing x′. (See[KMRT98,Proposition 2.22].)
Theproof incharacteristic not2can bewritten inasimilar way, orconcluded
from[CV13,Proposition 3.9]. (cid:3)
Proposition 3.2. Assume F is a 2-field and A is a division algebra of exponent 2
and degree 2n over F. If char(F) = 2 then for any two Artin-Schreier elements in
Athereexistsanelement,eitherArtin-Schreier orsquare-central, commutingwith
them both. Ifchar(F) , 2, for any twosquare-central elements in Athere exists a
square-central elementcommutingwiththemboth.
Proof. Assume char(F) = 2. Let x,t be two Artin-Schreier elements in A. Now,
t = t +t where t commutes with x and t satisfies xt +t x = t . The element
0 1 0 1 1 1 1
t2 + t = t2 + t t +t t +t2 + t + t decomposes similarly as T +T where T
0 1 0 0 1 1 0 1 0 1 0
commutes with x and T satisfies xT +T x = T . Clearly T = t2 +t2 +t and
1 1 1 1 0 0 1 0
T = t t +t t +t . However, t2 +t is central, which means that T = 0 and so
1 1 0 0 1 1 1
t t + t t = t . If t = 0 then x and t commute and the statement is trivial. If
1 0 0 1 1 1
t , 0 then F[x,t] = K[x,t ] where K is the field extension generated over F by
1 1
t2 and x+t . Clearly K[x,t ] is a quaternion algebra over K. If the K = F then
1 0 1
A = F[x,t]⊗A . NowA isofdegree2n−1andthereforeitcontainsafieldextension
0 0
of degree 2n−1 of the center, and since F is a 2-field it contains a quadratic field
extension of F (see[EKM08,Proposition 101.15]), whichisgenerated byeither a
square-central orArtin-Schreier element. Thiselement commutes with xand t. If
K , F thenitmustbeofdegreeapowerof2. Since F isa2-field, K mustcontain
a quadratic field extension of F which is generated by either a square-central or
Artin-Schreier element. Thiselementcommuteswith xandt.
Assume char(F) , 2. Let x,t be two square-central elements in A. Now,
t = t +t where t commutes with x and t anti-commutes with x. The element
0 1 0 1
t2 = t2+t t +t t +t2 decomposes similarlyasT +T whereT commuteswith
0 1 0 0 1 1 0 1 0
xandT satisfies xT = −T x. ClearlyT = t2+t2 andT = t t +t t . However,
1 1 1 0 0 1 1 1 0 0 1
t2 is central, which means that T = 0 and so t t = −t t . If t = 0 then x and t
1 1 0 0 1 1
commute and the statement is trivial. If t , 0 then F[x,t] = K[x,t ] where K is
1 1
the field generated over F by t2 and xt . Clearly K[x,t ] is a quaternion algebra
1 0 1
over K. If K = F then A = F[x,t]⊗A . Now A is of degree 2n−1 and therefore
0 0
it contains a field extension of degree 2n−1 of the center, and since F is a 2-field
itcontains aquadratic fieldextension of F,whichisgenerated byasquare-central
element. Thiselementcommutes with xandt. If K , F thenitmustbeofdegree
a power of 2. Since F is a 2-field, K must contain a quadratic field extension of
F which isgenerated byasquare-central element. Thiselement commutes with x
andt. (cid:3)
5
4. Fieldsofcohomological dimension2
The following fact appeared in [Kah90, Theorem 3] under the assumption
char(F) , 2:
Theorem 4.1. Let F be a field with cd (F) ≤ 2. For any form f ∈ I2F, E(f)
2 q
is a division algebra if and only if f is anisotropic. As a result, every algebra of
exponent 2 and power 2m is isomorphic to E(f) for some f ∈ I2F, and therefore
q
decomposes asthetensorproductofquaternion algebras.
Proof. By the remark preceding this theorem, we can assume char(F) = 2. First
weshowthateveryfrom f ∈ I2F isuniversal (i.e. represents everyelementof F):
q
Let D(f) denote the set of nonzero elements represented by f. Since cd (F) ≤ 2
2
and f ⊥ bf ∈ I3F for any b ∈ F×, wehave f ≃ bf. Thismeans F×D(f) ⊆ D(F)
q
andso D(f) = F×.
Nowweshowthatif f ∈ I2F isananisotropicformofdimensionatleast6and
q
f is isotropic for some K = F[x : x2 + x = a] then f = ˆ ⊥ f where ˆ is a
K K 0
hyperbolic planeand f isanisotropic: Since f isisotropic, f = (f′⊗[1,a]) ⊥ f
0 K 0
forsomebilinearform f′andanisotropicform f (see[EKM08,Proposition34.8]).
0
The dimension of f′ is equal to the Witt index of f . If the dimension of f′ is
K
greater than 1 then it contains a subform φ of dimension 2, and then φ ⊗ [1,a]
is a proper subform of f. This proper subform is in I2F, which means that it is
q
universal, andso f isisotropic, contradiction.
Let f beaforminI2F. Clearlyif f isisotropicthenE(f)isnotadivisionalge-
q
bra. Thereforeassumethat f isanisotropic. WewanttoshowthatE(f)isadivision
algebra. AccordingtoSpringer’sclassicaltheorem(see[EKM08,Corollary18.5])
f remains anisotropic under odd degree extensions. Therefore fF′ is anisotropic
where F′ istheodd closure of F. Since E(fF′) = E(f)⊗F′, itisenough toprove
that E(fF′)isadivisionalgebra. Assume F isa2-fieldthen.
If E(f) is split then since I2F (cid:27) Br (F), f is hyperbolic. Assume then that
q 2
E(f)isnonsplit. Hencethedimensionof f isatleast4. If f isofdimension4then
itisknownthat f isthenormformofthealgebraE(f),whichisaquaternionalge-
bra, and according to [EKM08, Corollary 12.5] a quaternion algebra is a division
algebraifandonlyifitsnormformisanisotropic. Thereforeassumethedimension
of f is2m ≥ 6.
We use induction on m. Since F is a 2-field with cd (F) ≤ 2, any finite field
2
extension K/F is also a 2-field with cd (K) ≤ 2. The induction hypothesis is that
2
foreach2 ≤ t < mandevery2-field Lwithcd (L) ≤ 2,ifφisananisotropic form
2
in I2Lofdimension 2t then E(φ)isadivision algebra. Since E(f)isnonsplit, itis
q
a matrix algebra over a nontrivial division algebra of degree a power of 2 over F.
ThisalgebracontainsanontrivialfieldextensionofF,andsinceF isa2-field,this
6
fieldextensioncontainsaquadraticfieldextension K ofF. Sincethedimensionof
f isatleast6,thedimensionoftheanisotropicpartof f is2(m−1). Now,Kisalso
K
a 2-field with cd (K) ≤ 2, so by the induction hypothesis, E(f )is of index 2m−2.
2 K
Ontheotherhand,theindexofE(f)isatleasttwicetheindexofE(f)⊗K = E(f ),
K
whichmeansthatitmustbe2m−1. Consequently, E(f)isadivision algebra. (cid:3)
Thefollowingtheoremextends[Bar14,Theorem3.3]whichpresentsasimilar
decomposition on the symbol-level fordivision algebras of exponent 2 overfields
ofcohomological 2-dimension 2andcharacteristic not2:
Theorem 4.2. Let F be afield with cd (F) ≤ 2, and Abe adivision algebra over
2
F of exponent 2 and degree 2n for some n ≥ 2. Let x and x′ be two commuting
elements in A with F[x] , F[x′]. If char(F) = 2 then assume x is either Artin-
Schreier or square-central and x′ is Artin-Schreier. If char(F) , 2 assume that
they are both square-central elements. Then A = Q ⊗ Q ⊗ ··· ⊗ Q such that
1 2 n
x ∈ Q and x′ ∈ Q .
1 2
Proof. Let B be the centralizer of F[x,x′] in A. It is enough to prove that B =
F[x,x′]⊗B′ forsomecentral simplealgebra B′ over F,becausethen A = A ⊗B′
0
(see [KMRT98, Theorem 1.5]) where A = C (B) is a degree 4 central simple
0 A
algebraofexponent≤ 2containing xandx′andthereforeitdecomposesasQ ⊗Q
1 2
such that x ∈ Q and x′ ∈ Q according to Lemma 3.1, and B′ decomposes as
1 2
Q ⊗···⊗Q according toTheorem4.1.
3 n
Let f be the unique quadratic form in I2F with E(f) (cid:27) A. In particular, f is
q
anisotropic. However, f isisotropic,because A⊗F[x]isnotadivisionalgebra.
F[x]
Since quadratic extensions are excellent, there exists g ∈ I F such that g is
q F[x]
isometric to the anisotropic part of f , and according to Lemma 2.1 we can
F[x]
assumeg∈ I2F.
q
Now,E(gF[x])=CA(F[x])andgF[x]isanisotropic. HowevergF[x,x′]isisotropic.
Assume char(F) = 2. Then g = φ ⊥ d[1,x′2 + x′] for some d ∈ F[x] and
F[x]
φ ∈ IqF[x] such that φF[x,x′] is the anisotropic part of gF[x,x′]. Since cd2(F) ≤ 2,
g ⊥ dg ishyperbolic,whichmeansthatg ≃ dg ,andhencewecanas-
F[x] F[x] F[x] F[x]
sumed = 1. Thereforeφisisometrictotheanisotropicpartofg ⊥ [1,x′2+x′].
F[x]
Since g ⊥ [1,x′2 + x′] is in I F and F[x]/F is excellent, there exists φ′ in I F
q q
such that φ′ is isometric to φ. Now, φ′ is in I2F[x,x′], and therefore ac-
F[x] F[x,x′] q
cordingtoLemma2.1thereexistsτinIq2F suchthatτF[x,x′] isisometrictoφ′F[x,x′].
Consequently, Bisarestriction of E(τ)to F[x,x′].
Assume char(F) , 2. Theng = φ ⊥ dh1,−x′2ifor somed ∈ F[x]and φ ∈
F[x]
IqF[x]suchthatφF[x,x′]istheanisotropicpartofgF[x,x′]. Sincecd2(F) ≤ 2,gF[x] ⊥
−dg ishyperbolic, which means that g ≃ dg , and hence wecanassume
F[x] F[x] F[x]
d = 1. Therefore φ is isometric to the anisotropic part of g ⊥ h−1,x′2i. Since
F[x]
7
g ⊥ h−1,x′2iisinI FandF[x]/F isexcellent,thereexistsφ′inI Fsuchthatφ′
q q F[x]
is isometric toφ. Now, φ′ is in I2F[x,x′], and therefore according toLemma
F[x,x′] q
2.1 there exists τ in Iq2F such that τF[x,x′] is isometric to φ′F[x,x′]. Consequently, B
isarestriction of E(τ)to F[x,x′]. (cid:3)
Corollary 4.3. If char(F) = 2 then for every two commuting Artin-Schreier ele-
ments x,x′ ∈ Awith F[x] , F[x′]thereexistsasquare-central elementzsuchthat
xz+zx= x′z+zx′ = z. Ifchar(F) , 2,foreverytwocommutingsquare-central el-
ements x,x′ ∈ Athereexistsasquare-central elementzanti-commuting withthem
both.
Proof. Since A = Q ⊗ Q ⊗ ··· ⊗ Q where x ∈ Q and x′ ∈ Q , there exist
1 2 n 1 2
square-central elements y ∈ Q and y′ ∈ Q such that y2,y′2 ∈ F and xy+yx = y
1 2
and x′y′+y′x′ = y′ ifchar(F) = 2,andyx = −xyandy′x′ = −x′y′ ifchar(F) , 2.
Takez= yy′. (cid:3)
Corollary 4.4. If char(F) = 2, x ∈ A is square-central, x′ ∈ A is Artin-Schreier
and xx′ = x′x, then there exist a square-central element z and an Artin-Schreier
elementwsuchthatwx+ xw = xandwz+zw = z = x′z+zx′ = z.
Proof. Since A = Q ⊗Q ⊗···⊗ Q where x ∈ Q and x′ ∈ Q , there exists an
1 2 n 1 2
Artin-Schreier element w ∈ Q whichsatisfies wx+ xw = xand wx′ = x′w. From
1
hereonwecanapplyCorollary 4.3. (cid:3)
Fromnowonassumewefocuson2-fields.
Theorem4.5. LetF bea2-fieldwithcd (F) ≤ 2,andAbeadivisionalgebraover
2
F of exponent 2 and degree 2n for some n ≥ 2. If char(F) = 2 then for every two
Artin-Schreierelemetns xand x′inAthereexistseitherachain x,y ,x ,y ,x′such
1 1 2
that x isArtin-Schreier,y andy aresquare-centralandxy +y x = x y +y x =
1 1 2 1 1 1 1 1 1
y and x′y +y x′ = x y +y x = y ,orachain x,y ,x ,y ,x ,y ,x′ withsimilar
1 2 2 1 2 2 1 2 1 1 2 3 3
properties. Ifchar(F) , 2thenforeverytwosquare-central elements xand x′ inA
there exists achain ofsquare-central elements x = x ,x ,x ,x ,x = x′ such that
0 1 2 3 4
xixi+1 = −xi+1xi foreach0 ≤ i≤ 3.
Proof. Assume char(F) , 2. According to Lemma 3.2, there exists a square-
central element x which commutes with them both. According to Corollary 4.3,
2
there exists a square-central element x commuting with x and x , and a square-
1 2
centralelement x commutingwith x and x′.
3 2
The proof in characteristic 2issimilar, making useof Corollaries 4.3 and 4.4.
(cid:3)
8
We say that two quaternion algebras Q and Q′ over F share a common slot if
forsomea,b,c ∈ F,either Q = [a,b)and Q′ = [a,c)or Q = [a,b)and Q′ = [c,b)
ifchar(F) = 2,orQ = (a,b)andQ′ = (a,c)ifchar(F) , 2. Wesaythattwotensor
products of quaternion algebras ⊗n Q and ⊗n Q′ share a common slot if there
i=1 i i=1 i
existiand jsuchthat Q and Q′ shareacommonslot.
i j
Theorem 4.6. Let F be a 2-field with cd (F) ≤ 2. For every two isomorphic
2
tensor products of quaternion algebras over F, ⊗n Q and ⊗n Q′, there exists a
i=1 i i=1 i
chain ⊗n Q,⊗n Q′′,⊗n Q′′′,⊗n Q′ such that every two adjacent tensor prod-
i=1 i i=1 i i=1 i i=1 i
uctsshareacommonslot.
Proof. If ⊗n Q is not a division algebra then it is isomorphic to M (F)⊗ Q′′ ⊗
i=1 i 2 2
···⊗Q′′ for some quaternion algebras Q′′. If char(F) = 2, write Q = [a,b) and
n i 1
Q′ = [a′,b′) for some a,a′ ∈ F and b,b′ ∈ F×. Then M (F) is isomorphic to
1 2
both[a,1)and[a′,1). Ifchar(F) , 2,write Q = (a,b)and Q′ = (a′,b′)forsome
1 1
a,a′,b,b′ ∈ F×. Then M (F)isisomorphic toboth(a,1)and(a′,1). Therequired
2
chainisobtained asaresult.
Assume ⊗n Q is a division algebra. Write Q = F[x,y : x2 + x = a,y2 =
i=1 i 1
b,xy + yx = y] and Q = F[x′,y′ : x′2 + x′ = a′,y′2 = b′,x′y′ + y′x′ = y′] if
2
char(F) = 2, and Q = F[x,y : x2 = a,y2 = b,xy = −yx] and Q′ = F[x′,y′ :
1 1
x′2 = a,y′2 = b,x′y′ = −y′x′] if char(F) , 2. If char(F) = 2 then there exists
an element z, either square-central or Artin-Schreier, commuting with x and x′.
If char(F) , 2 there exists a square-central element z commuting with x and x′.
Therefore, according to Theorem 4.2, A is isomorphic to ⊗n Q′′ where x ∈ Q′′
i=1 i 1
andz ∈ Q′′,and Aisalsoisomorphic to⊗n Q′′′ where x′ ∈ Q′′′ andz∈ Q′′′. The
2 i=1 i 1 2
required chainisobtained asaresult. (cid:3)
Acknowledgements
We thank Jean-Pierre Tignol and the anonymous referee whose remarks im-
provedthequalityofthepaperconsiderably.
Bibliography
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