Constructive basic theory of central simple algebras
aa r X i v : . [ m a t h . R A ] F e b Constructive basic theory of central simple algebras
Thierry CoquandComputer Science and Engineering Department, University of Gothenburg, Sweden, [email protected]
Henri Lombardi and Stefan NeuwirthLaboratoire de math´ematiques de Besan¸con, Universit´e Bourgogne Franche-Comt´e,France, [email protected] , [email protected] February 26, 2021
Contents
Basic definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Some general results about algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Wedderburn’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Some consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 F
94 Dynamical methods 11
Splitting central simple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Involutions of central simple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Some results about quaternion algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15First step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16Second step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Third step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Constructive versions of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Introduction
Wedderburn’s theorem (Wedderburn 1908, Theorem 22; Albert 1939, Theorem 3.17; Curtis 1999, The-orem VI.2.1; Mines, Richman, and Ruitenburg 1988, Theorem IX.5.1), which states that any centralsimple algebra is a matrix algebra over a division algebra, is an early instance of a nonconstructive resultin algebra. While less famous than Hilbert’s basis theorem in this respect, it has also played an impor-tant role in the development of abstract structures in mathematics, such as the use of modules satisfying The standard quaternion algebra A over a field K is central simple. But if we cannot decide whether K contains aroot of X + 1, we cannot decide whether A is a division algebra or the matrix algebra M ( K ) (Mines, Richman, andRuitenburg 1988, Exercise IX.1.4). § VI.2). Although noneffec-tive, it is also a basic result in the theory of central simple algebras , and this might explain why thepresentation of this theory in a constructive framework (Mines, Richman, and Ruitenburg 1988, § IX.5)stops essentially at giving possible weaker versions of this result that are valid effectively. The maingoal of this paper is to show that using in an essential way ideas from dynamical algebra (Mannaa andCoquand 2013; Lombardi and Quitt´e 2015) we can give a dynamical version of Wedderburn’s theorem,and to use this version to establish constructively most of the basic results of the theory of central simplealgebras. For instance, we show in a constructive setting the Skolem-Noether theorem (Skolem 1927;Noether 1933) (Theorem 2.11), as well as the fact that a finite-dimensional algebra A over a field F iscentral and simple if, and only if, A ⊗ F A op is a matrix algebra (Theorem 2.6).We illustrate these results by giving, in Section 5, an elementary constructive proof of a theorem byBecher (2016) (Theorem 5.1) which is a consequence of a celebrated theorem by Merkurjev. This paper is written in Bishop’s style of constructive mathematics (Bishop 1967; Lombardi andQuitt´e 2015; Mines, Richman, and Ruitenburg 1988). Classical sources we have used are Albert (1939);Blanchard (1972); Gille and Szamuely (2006); Baer (1940); Becher (2004, 2016); ˇSemrl (2006); Scharlau(1985).
Basic definitions and examples
We say that a vector space V over a field is strictly finite when we know a finite basis of V .We need first to explain what we mean by a central simple algebra over a field F . By algebra wealways mean an associative unital nonzero algebra A . Such a F -algebra is given with a ring morphism F → A . The field F has to be discrete: every element is 0 or invertible. So the morphism F → A isinjective. We generally identify F with F A ⊆ A . Moreover, the algebra A will always be a strictly finitevector space over F , with a basis ( ε , . . . , ε m ), and we will always take ε to be the unit 1 = 1 A of thealgebra. The algebra is given by a multiplication table ε i ε j = P ml =1 c lij ε l ( c lij ∈ F ). The coefficients c lij are called the structure constants of A (w.r.t. the basis ( ε , . . . , ε m )). • By left ideal of A , we mean a sub- F -vector space of A with a given basis which is a left A -module. A left ideal is proper when it is = A . An element a ∈ A generates a left ideal Aa = { xa : x ∈ A } ,a right ideal aA = { ax : x ∈ A } , and an ideal h a i = AaA = { P ni =0 x i ay i : n ∈ N , x i , y i ∈ A } . • If B is a F -algebra, the commutative direct sum A ⊗ F B is constructed as the free left B -modulegenerated by ε , . . . , ε m with the multiplication table as above. In consequence, if ( η , . . . , η n ) is a F -basis of B , the formal products ε i η k form a basis of A ⊗ F B ,with ε i η k = η k ε i , and by F -bilinearity,if ε i ε j = X mp =1 c pij ε p and η k η l = X nq =1 d qkl η q , then ( ε i η k )( ε j η l ) = X mp =1 X nq =1 c pij d qkl ( ε p η q ) . If A = M q ( F ), then M q ( F ) ⊗ F B ≃ M q ( B ).As a F -vector space, A ⊗ F B is the tensor product of A and B ; we shall use the usual terminologysaying that the algebra A ⊗ F B is the tensor product of the algebras A and B . This implies that these results are valid in an arbitrary sheaf topos. It uses induction up to ω ω and so is not primitive recursive; one should be able to use the work Carlucci, Mainardi,and Rathjen 2019 for a proof-theoretic analysis of the argument. It seems that the proof of this theorem in the reference Merkurjev 2006 is already essentially constructive, but we leavethis for future work. In other words, we consider usual left ideals with an explicit F -basis, possibly ∅ . When A and B are F -algebras, a commutative direct sum of A and B is a solution of the following universal problem:to find a F -algebra C and two morphisms α : A → C and β : B → C such that α ( a ) β ( b ) = β ( b ) α ( a ) for all a, b ; moreover,when we have two morphisms ϕ : A → D and ψ : B → D such that ϕ ( a ) ψ ( b ) = ψ ( b ) ϕ ( a ) for all a, b , then there is aunique morphism θ : C → D such that θα = ϕ and θβ = ψ . A solution of this universal problem is C = A ⊗ F B withthe embeddings α : A → C, a a ⊗ B and β : B → C, b A ⊗ b , and the multiplication law in C is defined by( a ⊗ b ) · ( a ′ ⊗ b ′ ) = ( aa ′ ) ⊗ ( bb ′ ). By subalgebra B of an algebra A we intend a F -algebra with an injective morphism B → A ; if weidentify B with its image in A , we have 1 B = 1 A . The F -basis of A can be modified so that the F -basis of B is an initial subsequence of the F -basis of A . The algebra A has a natural structure of left B -module. When A is a free left B -module with basis ( c , . . . , c r ), we write r = dim B ( A ) = [ A : B ]and we have [ A : F ] = [ A : B ][ B : F ]. • If D is a subalgebra of an algebra C , the centraliser of D in C is the subalgebra { x ∈ C : ∀ y ∈ D, xy = yx } : given the constructive theory of vector spaces over a discrete field, we can computea basis of this centraliser. We denote it by Z C ( D ). The subalgebra Z C ( C ) = Z( C ) is called the center of C . • That A is central means that Z( A ) = F A : if xy = yx for all y then x = r A for some r in F . Inother words, if x / ∈ F A then we can find y such that xy = yx (because we can compute the kernelof the linear map y xy − yx ). • That A is simple can be expressed by stating that if a = 0 then AaA = A ; in other words we canfind x i ’s and y i ’s such that P i x i ay i = 1 A . Note that a simple commutative F -algebra K is a finitefield extension of F . • We use the following notation: End F ( A ) is the algebra of F -linear endomorphisms of A ; End A/F ( A )is the monoid of endomorphisms of the F -algebra A . The vector space End F ( A ) is isomorphic to M m ( F ) through the basis ( ε , . . . , ε m ). • We denote by L Aa or L a the F -linear endomorphism of A defined by L a x = ax . Similarly R Ab x =R b x = xb . These maps embed A and A op in End F ( A ). We have L a R b = R b L a . This provides anatural map A ⊗ F A op → End F ( A ) ≃ M m ( F ). When the F -basis of A is clear from the contextwe say that we consider the canonical map A ⊗ F A op → M m ( F ). • Note that if ϕ ∈ End F ( A ) satisfies ϕ L a = L a ϕ for all a ∈ A , then ϕ is R b with b = ϕ (1). So,with the embeddings of the previous item, A op is the centraliser of A in End F ( A ) and A is thecentraliser of A op in End F ( A ). • A is split iff it is isomorphic to a matrix algebra M q ( F ). It is splittable if there exists a commutative F -algebra K such that A K := K ⊗ F A is isomorphic to a matrix algebra M q ( K ). We say that K splits A . In classical mathematics, K can be taken to be a finite field extension. The constructivedefinition is equivalent in classical mathematics to the classical one. In fact, if we know a fieldquotient K/J = L of K , we have A L = L ⊗ F A ≃ L ⊗ K A K ≃ L ⊗ K M q ( K ) ≃ M q ( L ). Basic examples • A matrix algebra M q ( F ) is central simple. • If B is central simple, M q ( B ) is central simple. • An F -algebra D which is a division ring is simple. Its center K is a finite extension of F . In thiscase, the K -algebra D is called a division algebra .Recall also that M q ( M r ( C )) ≃ M qr ( C ). Some general results about algebras
The next lemma is similar to the description of linearly disjoint extensions of a field when A is acommutative field. We recall that our algebras are all nonzero by definition. Lemma 1.1.
Let A be an arbitrary F -algebra, B and C commuting subalgebras of A , and ϕ : B ⊗ F C → A the natural morphism. Let ( b i ) i be a F -basis of B . The following are equivalent.1. ϕ is an isomorphism.2. ϕ is onto and the b i ’s are C -linearly independent. In other words, we consider usual subalgebras which are strictly finite over F . . ϕ is onto and dim F ( B ) dim F ( C ) = dim F ( A ) .4. ϕ is injective and dim F ( B ) dim F ( C ) = dim F ( A ) .If this is the case we write A = B ⊗ F C . Lemma 1.2.
Let A , B , D be arbitrary F -algebras and ϕ : A → B a morphism of algebras. Let ϕ = ϕ ⊗ F D : A ⊗ F D → B ⊗ F D . Then ϕ is injective (resp. surjective, bijective) if, and only if, ϕ isinjective (resp. surjective, bijective).Proof. If D ≃ F r ( r >
0) as a vector space, then ϕ ≃ ⊕ ri =1 ϕ . Lemma 1.3.
Let A , B be F -algebras, C and D subalgebras, respectively, C ′ the centraliser of C in A ,and D ′ the centraliser of D in B . Then C ′ ⊗ F D ′ is the centraliser of C ⊗ F D in A ⊗ F B : the naturalmap Z A ( C ) ⊗ F Z B ( D ) → A ⊗ F B is injective and its image is equal to Z A ⊗ F B ( C ⊗ F D ) .Proof. It is clear that elements of C ′ ⊗ F D ′ commute with all elements of C ⊗ F D , so C ′ ⊗ F D ′ iscontained in Z A ⊗ F B ( C ⊗ F D ). The other inclusion is less obvious. We use a F -basis ( ε i ) mi =1 of A (with ε = 1 A ) and a F -basis ( η j ) pj =1 (with η = 1 B ) of B , where ( η j ) qj =1 is a F -basis of D ′ . Then weexamine when a vector u = P mi =1 ε i ⊗ y i commutes with all elements of C ⊗ F D . For d ∈ D this implies0 = u (1 A ⊗ d ) − (1 A ⊗ d ) u = P i ε i ⊗ ( y i d − dy i ), and since the ε i ’s are B -linearly independent, each y i belongs to D ′ . So we write y i = P qj =1 f ij η j ( f ij ∈ F ). We get u = P qj =1 x j ⊗ η j for some x j ’s in A .For c ∈ C we get 0 = u ( c ⊗ B ) − ( c ⊗ B ) u = P qj =1 ( x j c − cx j ) ⊗ η j , and since the η j ’s are A -linearlyindependent, each x j belongs to C ′ .We have the following corollary. Lemma 1.4.
Let A be a F -algebra and B a central F -algebra.1. Let C be a subalgebra of A and C ′ = Z A ( C ) . Then C ′ ⊗ F F is the centraliser of C ⊗ F B in A ⊗ F B .In particular, Z A ⊗ F B ( F ⊗ F B ) = A ⊗ F F .2. If A and B are central, then conversely Z A ⊗ F B ( A ⊗ F F ) = F ⊗ F B (a more subtle similar resultis given in Theorem 2.13). Lemma 1.5.
Let A be a F -algebra. For each a ∈ A there is an n ∈ N and an idempotent e ∈ F [ a ] suchthat ea n = a n and h e i = h a n i = h a n + r i for all r and • if e = 1 , then a is invertible; • if e = 0 , then a is nilpotent; • in the remaining case, AeA is a nonzero proper ideal of A .In particular, a ∈ A is left regular if, and only if, it is left invertible if, and only if, it is invertible.Proof (Lombardi and Quitt´e 2015, Lemma VI-3.13). In the commutative algebra F [ a ] we write the min-imal polynomial g of a as uX n (1 − Xf ( X )) with u ∈ F × and f ∈ F [ X ]. Then a n (1 − ac ) = 0 (where c = f ( a ) ∈ F [ a ]), so a n = a n +1 c = a n c n and e = ( ac ) n is the idempotent such that ea n = a n and h e i = h a n i .Note that e = 0 iff g = X n and e = 1 iff g (0) = 0. Lemma 1.6.
Let V be a F -vector space of dimension r , A = End F ( V ) ≃ M r ( F ) , and L a subalgebraof A which is a field. Let ℓ := [ L : F ] . Then V is a free L -vector space with an L -basis ( x , x , . . . , x q ) ,and r = ℓq . Moreover B := Z A ( L ) ≃ M q ( L ) and Z A ( B ) = L .Proof. There is a natural structure of L -vector space on V given by L × V → V, ( S, x ) Sx = S ( x ).Let x = 0 ∈ V : the sub- L -vector space of V generated by x is Lx , F -isomorphic to F ℓ . We can constructan L -basis ( x , . . . , x q ) of V , so r = [ V : F ] = [ V : L ][ L : F ] = ℓq . We have End L ( V ) ⊆ End F ( V ).An element T ∈ End F ( V ) commutes with all S ∈ L if, and only if, it belongs to End L ( V ) ≃ M q ( L ). Inother words B := Z A ( L ) = End L ( V ). A priori, the centraliser C = Z A ( B ) contains L . Since L ⊆ B ,the elements of C commute with all elements of L , i.e. C ⊆ B = End L ( V ), and since L is the center ofEnd L ( V ), we get C ⊆ L . We identify F with the subalgebra F I r of A , so that F ⊆ L . This basis can be extracted from a F -basis of V .
4e give a generalisation of this lemma in Theorem 3.6.An invertible element u of a F -algebra A defines an inner F -automorphism of A , denoted by Int( u )and defined by Int( u )( a ) = uau − for all a ∈ A . Proposition 1.7.
Let σ be a F -automorphism of M q ( F ) . There is an u in GL q ( F ) such that σ = Int( u ) . In fact, u is well-defined up to a unit of F since uau − = vav − for all a implies that v − u is in thecenter F I q of M q ( F ). So the group of automorphisms of M q ( F ) is isomorphic to PGL q ( F ). Proof.
We follow the elementary proof in ˇSemrl 2006. Let F q denote the vector space of column vectors.Let y , y, z in F q such that σ ( y y T ) z = 0. We define u ∈ End F ( F q ) = M q ( F ) by ux = σ ( xy T ) z for x ∈ F q . We have for each a in M q ( F ) uax = σ ( axy T ) z = σ ( a ) σ ( xy T ) z = σ ( a ) ux for all x ∈ F q and so ua = σ ( a ) u . Furthermore uy = σ ( y y T ) z = 0. Hence for an arbitrary t ∈ F q we can find anelement c ∈ GL q ( F ) such that cuy = t . Letting b = σ − ( c ) we get uby = σ ( b ) uy = cuy = t andhence u is surjective. So u ∈ GL q ( F ) and σ ( a ) = uau − for all a ∈ M q ( F ). Remark . The automorphism group of GL n ( K ) and other “classical groups” for a (skew) field K is aclassical topic. See e.g. Dieudonn´e 1971. Wedderburn proved that any simple algebra is isomorphic to M q ( D ) for a suitable division algebra D .The proof uses the law of excluded middle and we shall prove constructively a dynamical form of thisresult (Theorem 2.4).An important consequence of Wedderburn’s theorem is that an algebra A is central simple if, and onlyif, A ⊗ F A op is a matrix algebra (Theorem 2.6 and Proposition 2.7). Also, the important corollaries 2.8and 2.9 seem to have no simple elementary proof. Wedderburn’s theorem
Proposition 2.1. If A is simple and has a nonzero proper left ideal L , then L contains a nontrivialidempotent.Remark. The hypothesis means simply that we have a nonzero a ∈ A such that Aa = A . The conclusionwill be that Aa contains a nonzero idempotent. Proof.
We can assume that L = Aa = A with a = 0. Lemma 1.5 gives an idempotent e ∈ Aa and itremains to consider the case where e = 0 and a is nilpotent. We can assume that a = 0 and a = 0.Since A is simple, we write P i y i az i = 1, so (cid:16) X i y i az i (cid:17) = X ik y i az i y k az k = X ik y i av ik az k = 1 . So there is a b = v ik such that a = aba = 0. Note that a = 0. If Aa = Aa , we replace a by a with Aa strictly contained in Aa and we are done by induction. So we may assume that Aa = Aa . We havean x ∈ A such that xaba = a . Let a = ba and e = bxa : a = 0 since aa = a = 0, ea = a , e = 1since e ∈ Aa , and e = 0 since ea = a = 0. We have ( e − e ) a = 0. If e − e = 0 we are done. If y = e − e = 0 then Ay is a nonzero sub-left ideal of Aa and it is proper since Aya = 0 and Aaa = 0.So we can replace a with e − e and we are done by induction.The next result (Theorem 2.4) is a constructive/dynamical way to formulate the classical fact thatif A is simple then A can be written M q ( F ) ⊗ F B ≃ M q ( B ) where B is a division algebra (i.e. the classicalformulation of Wedderburn’s theorem). Let us start with an essential remark. In Gonthier 2011, the problem with non-constructiveness is solved by using a double-negation translation. It would beinteresting to connect these two a priori different methods, but we leave this for future work. emma 2.2. Let A be a simple F -algebra. If e is a nontrivial idempotent of A , then eAe is again asimple F -algebra with e as unit element (note that it is not a subalgebra of A ).Proof. If we let A ′ = eAe = { x ∈ A : x = exe } , then A ′ is stable by addition and multiplication,and e is a unit element in A ′ . Moreover if x = 0 in A ′ we write P i y i xz i = 1 in A , which gives P i ( ey i e ) x ( ez i e ) = P i e ( y i xz i ) e = e in A ′ .Furthermore eAe ( A if e = 1. Note that letting f = 1 − e we get a direct sum of F -subspaces A = eAe ⊕ eAf ⊕ f Ae ⊕ f Af . By Proposition 2.1, if we find a noninvertible element in eAe then we cansplit e into smaller idempotents.If A = M q ( B ) for a F -algebra B , let us use the elementary matrices E ij ∈ M q ( B ). We have E ii idempotent and P i E ii = 1. Also E ij E jl = E il and E ij E kl = 0 if j = k . The F -algebras E ii AE ii areisomorphic to B . We may state a converse result, defining a matrix algebra decomposition for A as asequence of nonzero elements e ij (1 ≤ i, j ≤ q ) such that: • e ij e jl = e il for all i, j, ℓ , • e ij e kl = 0 if j = k , • e + · · · + e q (where e i denotes e ii ). Lemma 2.3 (matrix algebra decomposition) . Assume that A has a matrix algebra decomposition ( e ij ) ≤ i,j ≤ q and let A i = e i Ae i . Then A ≃ A i foreach i and A ≃ M q ( A ) .Proof. Elements of A i are characterised by e i xe i = x . Let ψ ij : A j → A i be defined by ψ ij ( x ) = e ij xe ji .Elementary computations show that ψ ij is a F -algebra morphism and that ψ ij ◦ ψ ji = Id A i .For a ∈ A let a ij = e i ae j . We have a ij = e a ij e ∈ A . We note that M q ( A ) has a natural structureof free A -bimodule of rank q and we define ϕ : A → M q ( A ) by ϕ ( a ) = ( a ij ) = X i,j a ij E ij = X i,j E ij a ij . We claim that ϕ is a F -algebra isomorphism. Clearly ϕ is linear and ϕ ( e kℓ ) = E kℓ . So ϕ (1) = ϕ ( P k e k )is the identity matrix in M q ( A ). An elementary computation shows ϕ ( ab ) = ϕ ( a ) ϕ ( b ). If ϕ ( a ) = 0, weget e i ae j = e i a ij e j = 0 for all i, j , hence a = P i,j e i ae j = 0. So ϕ is an isomorphism. Theorem 2.4 (Wedderburn’s theorem, constructive form) . If A is simple and contains a nonzero non-invertible element, then A can be written M q ( F ) ⊗ B ≃ M q ( B ) , where q > and B is simple. Moreoverif A is central simple, then so is B . Note that q [ B : F ] = [ A : F ]. Proof.
By Lemma 2.3, it is enough to prove that A admits a matrix algebra decomposition. By Propo-sition 2.1, A contains a nontrivial idempotent e . We start with the orthogonal nontrivial idempotents e, f where e + f = 1 A and we refine this decomposition ( e, f ) of 1 A into a finer one ( e , . . . , e n ) as longas we don’t get a matrix algebra decomposition.Dynamically we do as if e Ae is a division algebra, and as if we cannot split further any e i for i >
1. Indoing this, we follow the argument of Albert 1939, Theorem 3.9.We define A ij = e i Ae j . We have A ij A jl = e i Ae j e j Ae l = e i Ae l = A il ( Ae j A = A since A is simple) and A ij A kl = 0 if j = k .For each i > A i A i = A . So we can find e i in A i and a i ∈ A i such that a i e i = a i = 0in A . If A a i = A , Proposition 2.1 gives a nontrivial idempotent in A which refines our decom-position. So we can assume A a i = A , and a i invertible in A by Lemma 1.5. Let b i ∈ A with b i a i = a i b i = e , and let e i = b i a i ∈ A i ; we have e i e i = e . This is done for all i > e = e and e ij = e i e j for i, j >
1; we have e ij e jl = e il for all i, j, l and e ij e kl = 0 if k = j . Note that e ii = e i e i is nonzero since e ii e i = e i = 0. So e ii is a nonzero idempotent in A ii . If e ii = e i we can split further e i . So we can assume e i = e ii and we have a matrix algebra decompositionfor A . Let us check that A is simple. Let x = 0 in A ; we have Y ℓ , Z ℓ ∈ A such that P ℓ Y ℓ xZ ℓ = 1 A . So Here E ij is seen as an element of M q ( B ). So if E ′ ij is the corresponding elementary matrix in M q ( F ), then E ij = E ′ ij ⊗ B . ,
1) w.r.t. to the matrix algebra decomposition M q ( A ) ≃ A , weget an equality P r y r xz r = 1 A . Finally, if x ∈ A commutes with all elements of A , then x M q ( A ) commutes with all matrices in M q ( A ), so A is central if A is central. And A is the central simple F -algebra B we are looking for.In classical mathematics, we deduce from Theorem 2.4 that we can force B to be a division algebrasince otherwise the theorem works for B , which can be written in the form M r ( C ) with r > F -algebras, telling us whether there is a nonzerononinvertible element, we get the classical form of Wedderburn’s theorem. Some consequences
Proposition 2.5. If A is central and the canonical map A ⊗ F A op → M m ( F ) is not an isomorphism,then we can find a nonzero noninvertible element in A . This is a key result, and it appears in some form in all treatments that we have seen (Albert 1939calls it a “fundamental” lemma and notices that the proof relies on the first basic results of the theoryonly).
Proof.
If this map is not an isomorphism, this means that the m elements L ε i R ε j are not free in M m ( F ).There is then a nontrivial combination P L a j R ε j = 0 with at least one a j = 0. If all a j = 0 are invertiblethen we can assume one a j to be 1 A . We have P a j xε j = 0 for all x . If all a j were in the center F ε wewould get that the ε i are linearly dependent. So there is a j = j and a y ∈ A such that a j y = ya j .We then have P j a ′ j xε j = P j ( a j y − ya j ) xε j = 0 for all x . This nontrivial linear combination has asmaller support: it has less nonzero elements since a ′ j = 0. We keep doing this until we find a nonzerononinvertible element. Theorem 2.6. If A is central simple then A ⊗ F A op is a matrix algebra. More precisely, the canonicalmap A ⊗ F A op → M m ( F ) is an isomorphism.Proof. This holds by induction on the dimension of A using the previous results. More precisely, if thecanonical map A ⊗ F A op → M m ( F ) is not an isomorphism, we can find a nonzero noninvertible elementin A and we can write A = M q ( B ) where B is central simple and q > B ⊗ F B op ≃ M r ( F ), so A ⊗ F A op ≃ M qr ( F ). Proposition 2.7. If A ⊗ F B is central simple then both A and B are central simple. In particular, if A ⊗ F A op is a matrix algebra then A is central simple.Proof. Let ( ε , . . . , ε m ) and ( η , . . . , η p ) be F -bases of A and B , with ε = 1 A , η = 1 B . The elements ε i ⊗ η j form a basis of C = A ⊗ F B . Any element of C can be written as P j a j ⊗ η j with the a j ’sin A . If x = 0 in A , from 1 C ∈ C ( x ⊗ B ) C we deduce 1 A ∈ AxA : inspect the equality ( P j a j ⊗ η j )( x ⊗ B )( P j a ′ j ⊗ η j ) = 1 C . So A is simple. In a similar way, A is central (or use Lemma 1.3). Note.
From Theorem 2.6 and Proposition 2.7 we get the following important characterisation: A iscentral simple if, and only if, the canonical map A ⊗ F A op → M m ( F ) is an isomorphism.In the following corollary, the field K containing F is not supposed to be finitely generated over F . Corollary 2.8.
Let A be a F -algebra, K a discrete field extension of F and A K = K ⊗ F A . Then A is F -central simple if, and only if, A K is K -central simple.Proof. 1. Assume that A is central simple. The K -algebra A K ⊗ K A Kop ≃ K ⊗ F ( A ⊗ F A op ) is isomorphicto M m ( K ) since A ⊗ F A op ≃ M m ( F ) by Theorem 2.6. Hence A K is central simple by Proposition 2.7. Assume that A K is central simple. First we check that A is simple. If B = A K is simple and0 = x ∈ A , we have to see that ε ∈ AxA = P j,k F ε k xε j . This can be seen as a solution of a F -linear system whose coefficients are given by the coordinates of x and the structure constants of A . But ε ∈ BxB = P j,k Kε k xε j . Since this F -linear system has a solution in K , it has a solution in F .Now we check that A is central. For a ∈ F , let δ a : A → A be defined by δ a = L a − R a . The center We can also use Lemma 1.3. A ) is the intersection of the kernels Ker( δ ε i ). This is a linear algebra computation, which implies thatZ( A K ) is obtained from Z( A ) by scalar extension from F to K . Since Z( A K ) = ε K , this implies thatZ( A ) = ε F . Corollary 2.9.
Let A and B be central simple algebras. Then A ⊗ F B is central simple.Proof. We have ( A ⊗ F B ) ⊗ F ( A ⊗ F B ) op ≃ ( A ⊗ F A op ) ⊗ F ( B ⊗ F B op ) ≃ M p ( F ) ⊗ F M q ( F ) ≃ M pq ( F ).We conclude by Proposition 2.7. Remark . From Proposition 2.7 and Corollary 2.9 we get that A ⊗ F B is central simple if, and onlyif, A and B are central simple. We have also seen that A ⊗ F B is central if, and only if, A and B arecentral. If A is simple and B central simple, it is possible to show that A ⊗ F B is simple. Finally, if K is a finite field extension of F , K is simple, but for x ∈ K \ F , the element x ⊗ − ⊗ x is nonzerononinvertible in K ⊗ F K , so K ⊗ F K is not simple. Theorem 2.11 (Skolem-Noether) . If A is central simple and σ is a F -automorphism of A , then we canfind an invertible element w of A such that σ = Int( w ) .Proof. We follow the proof in Baer 1940, page 590. We use that A ⊗ F A op ≃ M q ( F ). One extends σ into an automorphism of M q ( F ) which leaves each element R b invariant. By Proposition 1.7, thisautomorphism is given by an element u of GL q ( F ). But then u commutes with each R b , hence is of theform L a , and σ is the inner automorphism given by a .The next theorem can be seen as detailing Lemma 2.3 when A is central simple. Theorem 2.12 (Albert 1939, Theorem 1.17) . Let B ≃ M q ( F ) be a matrix subalgebra of a central simplealgebra A . Then A = B ⊗ F C ≃ M q ( C ) with C = Z A ( B ) . Proof.
Let ( e ij ) ≤ i,j ≤ q be a matrix algebra decomposition for B . For a ∈ A write a ij = P k e ki ae jk . Onehas a ij e rs = e ri ae js = e rs a ij . So the a ij ’s belong to C . Moreover X ij a ij e ij = X ijk e ki ae jk e ij = X ij e ii ae jj = (cid:0) X i e ii (cid:1) a (cid:0) X j e jj (cid:1) = a, so the natural morphism ϕ : B ⊗ F C → A is onto. By Lemma 1.1 it is sufficient to prove that the e ij ’sare C -linearly independent. So consider c ij ’s in C and assume c = P ij c ij e ij = 0. Then e pr ce sp = 0 and0 = e pr ce sp = e pr (cid:0) X ij c ij e ij (cid:1) e sp = c rs e pp , hence c rs = P p c rs e pp = 0 for all r, s .Now we generalise Theorem 2.12 without assuming that B is split. Theorem 2.13 (double centraliser theorem) . Let A be a central simple algebra and B a central simplesubalgebra. Then A = B ⊗ F C where C = Z A ( B ) . Consequently [ A : F ] = [ B : F ][ C : F ] , C is centralsimple and B = Z A ( C ) .Proof. Let us consider the central simple algebra A ⊗ F B op that contains B ⊗ F B op ≃ M q ( F ) as subal-gebra. By Lemma 1.4 (with B op instead of B ), C ⊗ F F is the centraliser of B ⊗ F B op in A ⊗ F B op . So A ⊗ F B op = C ⊗ F B ⊗ F B op by Theorem 2.12. By Lemma 1.2 (with D = B op ), we get A = C ⊗ F B .The fact that [ A : F ] = [ B : F ][ C : F ] is an immediate consequence of A = B ⊗ F C . The fact that C is central simple follows from Proposition 2.7. Finally, since B and C are central, B ⊗ F F and F ⊗ F C centralise each other in B ⊗ F C (see Lemma 1.4). Recall that when B and C are commuting subalgebras of a given algebra A , the notation A = B ⊗ F C means that thenatural morphism B ⊗ F C → A is an isomorphism (Lemma 1.1). The Brauer group of a discrete field F Definition 3.1.
Two central simple F -algebras A and B are said to be Brauer equivalent if there are m and n such that M m ( A ) ≃ M n ( B ). We denote this by A ∼ F B or [ A ] Br F = [ B ] Br F or [ A ] = [ B ].NB. It is easy to see that • ∼ F • is an equivalence relation by using M p ( M q ( A )) ≃ M pq ( A ). Lemma 3.2. If A ∼ F A ′ and B ∼ F B ′ then A ⊗ F A ′ ∼ F B ⊗ F B ′ Proof.
We have M m ( A ) ≃ M n ( A ′ ) and M p ( B ) ≃ M q ( B ′ ). Since M m ( A ) ≃ M m ( F ) ⊗ F A we get M mp ( A ⊗ F B ) ≃ M m ( A ) ⊗ F M p ( B ) ≃ M n ( A ′ ) ⊗ F M q ( B ′ ) ≃ M nq ( A ′ ⊗ F B ′ ) . As a corollary, the equivalence classes of central simple algebras form a group with neutral element1 = [ F ], the inverse of [ A ] being [ A op ] by Theorem 2.6. This is called the Brauer group of F , denotedby Br( F ).Note the following simplification rule: if A , B , C are central simple algebras such that A ⊗ F C ∼ F B ⊗ F C , this can be written [ A ] F [ C ] F = [ B ] F [ C ] F , so [ A ] F = [ B ] F , i.e. A ∼ F B .The next theorem is a generalisation of Proposition 1.7: when we take A = B = F and two isomor-phisms M m ( A ) → C and M m ( B ) → C , we get Proposition 1.7 since the proof ends at the first step ( F isa division algebra over itself!). It should be interesting to compare the concrete computations involvedin the two proofs. Note that σ ( y y T ) z = 0 in the proof of Proposition 1.7 is similar to d pq = 0 in theproof of Theorem 3.3. Theorem 3.3.
Let A , B be central simple algebras.1. If M m ( A ) ≃ C ≃ M n ( B ) then there exist m ′ , A ′ , n ′ , B ′ such that A ≃ M m ′ ( A ′ ) , B ≃ M n ′ ( B ′ ) , mm ′ = nn ′ and A ′ ≃ B ′ .More precisely, consider the matrix algebra decompositions ( e ij ) ≤ i,j ≤ mm ′ and ( f ij ) ≤ i,j ≤ nn ′ in C corresponding respectively to isomorphisms C ≃ M mm ′ ( A ′ ) and C ≃ M nn ′ ( B ′ ) . Then there is aninvertible g ∈ C such that f ij = ge ij g − for all i, j . In particular, the automorphism x gxg − of C sends e Ce , which is isomorphic to A ′ , to f Cf , which is isomorphic to B ′ .2. In particular,(a) [ A ] F = 1 if, and only if, A is split.(b) If A ∼ F B and [ B : F ] = [ A : F ] then A ≃ F B .(c) Assume that A = M r ( D ) where D is a division algebra. Then B ∼ F A if, and only if, thereexists an s such that B ≃ M s ( D ) , and B ≃ A if, and only if, B ∼ F A and r = s . Note that when A and B are division algebras, the theorem implies that m ′ = n ′ = 1, m = n and A ≃ B . This is the classical form of Theorem 3.3 in the classical litterature. We will follow the proofgiven in this case in Albert 1939, Theorem 4.1. Proof. 1 . The isomorphism ϕ A : C → M m ( A ) is given by a matrix algebra decomposition ( e ij ) ≤ i,j ≤ m for C and an isomorphism A ∼ −→ e Ce . Identifying A with e Ce , we get ϕ A ( x ) = P i,j x ij E ij = P i,j E ij x ij , where x ij = e i xe j ∈ A and the E ij ’s are the usual elementary matrices in M m ( A ). Similarlythe isomorphism ϕ B : C → M n ( B ) is given by a matrix algebra decomposition ( f ij ) ≤ i,j ≤ n for C and anisomorphism B ∼ −→ f Cf . Identifying B with f Cf , we get ϕ B ( y ) = P i,j y ij F ij = P i,j F ij y ij , where y ij = f i xf j ∈ B and the F ij ’s are the usual elementary matrices in M n ( B ).In a first step we assume A and B to be division algebras, and m ≤ n . In the following summations,all i, j are in { , . . . , m } . We write ϕ A ( f ) = P i,j d ij E ij with the d ij ’s in A . Some d pq is nonzero, henceinvertible.We let a = d − pq e p f and b = f e q . We have af b = ab ; since f = f we get ab = e by computing ϕ A ( ab ) = d − pq E p ϕ A ( f ) E q = d − pq X i,j E p d ij E ij E q = d − pq X i,j d ij E p E ij E q = d − pq d pq E = ϕ A ( e ) . We have ba = f e q d − pq e p f ∈ B and ( ba ) = b ( ab ) a = be a = f e q e a = f e q a = ba . So ba is anonzero idempotent in B , i.e. ba = f . We let (recall that m ≤ n ) h = X mi =1 e i af i , g = X mj =1 f j be j (1)9e obtain hg = X i,j e i af i f j be j = X i e i af be i = X i e i e e i = X i e i e i = 1 . So gh = hg = 1. Next, since be a = baba = ba = f , ge pr g − = ge pr h = X i,j f j be j e pr e i af i = f p be af r = f p f f r = f pr . Now 1 = P mi =1 e i , so 1 = g (cid:0) P mi =1 e i (cid:1) g − = P mi =1 f i , which implies m = n .Let us now consider the general case. We assume m ≤ n . We have used twice the hypothesis that A and B are division algebras: the first time when saying that d pq is invertible in A , the second time whensaying that the nonzero idempotent ba in B = f Cf is equal to f . Let us rewrite our proof in the moregeneral context. If we find that d pq is not invertible, we know by Theorem 2.4 that A is isomorphic tosome M m ′ ( A ′ ) with m ′ > A ′ : F ] < [ A : F ]. So we can replace in our hypothesis A by A ′ and m by mm ′ . If mm ′ ≥ n , we permute A ′ and B . In this case, a similar problem can appear with the new d pq in B . This kind of problem can only happen a finite number of times. So we assume now m ≤ n and d pq invertible in A . If the nonzero idempotent ba in f Cf is not equal to f , we know that B is isomorphicto some M n ′ ( B ′ ) with n ′ > B ′ : F ] < [ B : F ]. So we can replace in our hypothesis B by B ′ and n by nn ′ . And all computations work out well. . Left to the reader. Lemma 3.4.
Let A and B be central simple F -algebras, and K an extension of F . If A ∼ F B , then A K ∼ K B K . In particular, if A ∼ F B , then K splits A if, and only if, K splits B .Proof. If M m ( A ) ≃ M n ( B ) as F -algebras, then M m ( A K ) ≃ K ⊗ F M m ( A ) ≃ K ⊗ F M n ( B ) ≃ M n ( B K )as K -algebras.This gives a natural group morphism Br( F ) → Br( K ).The next theorem is a generalisation of Lemma 1.6. We begin with an easy lemma. Lemma 3.5.
Let A be a F -algebra, L ⊇ F a commutative subalgebra, and B = Z A ( L ) . Then L ⊆ B and L is maximal among the commutative subalgebras of A if, and only if, B = L .Proof. If M ⊇ L is commutative, then M ⊆ B . If x ∈ B \ L , then L [ x ] is commutative and strictlycontains L . Theorem 3.6 (Albert 1939, Theorem 4.12) . Let A be a central simple F -algebra, L ⊇ F a finitelygenerated subfield of A , and B = Z A ( L ) in A . Then B is L -central simple, B ∼ L A L = L ⊗ F A , and wehave [ B : L ][ L : F ] = [ A : F ] . In particular, a subfield L of A cannot have [ L : F ] > [ A : F ] , and if [ L : F ] = [ A : F ] then B = L and L is maximal.Note. With Definition 4.2 of deg F ( A ), the equality [ B : L ][ L : F ] = [ A : F ] reads deg L ( B )[ L : F ] =deg F ( A ). Proof.
Let ℓ = [ L : F ] and m = [ A : F ]. We consider D = A ⊗ F A op ≃ M m ( F ). We note that L ⊗ F F (which is isomorphic to L ) is a subfield of D . So, by Lemma 1.6, the centraliser E = Z D ( L ⊗ F F )is isomorphic to M q ( L ), where q = [ D : L ] and ℓq = m . Moreover, E = B ⊗ F A op by Lemma 1.3.The algebras E and B have a natural structure of L -algebra and E is L -central simple. The algebra E contains L ⊗ F A op ≃ A opL which is L -central simple by Corollary 2.8. Then, by Theorem 2.13, we have E = ( L ⊗ F A op ) ⊗ L C , where C is the centraliser of L ⊗ F A op in E = B ⊗ F A op , and C is L -centralsimple. By Lemma 1.3, C = B ⊗ F F ≃ B . So B is L -central simple. We have A L ⊗ L A opL ≃ D L ≃ M m ( L ) ∼ L M q ( L ) ≃ E ≃ C ⊗ L A opL , which implies C ∼ L A L .Finally, [ E : F ] = dim F ( M q ( L )) = q ℓ = mq , so E = B ⊗ F A op implies [ B : F ] = q , so that [ B : L ] = q/ℓ .So the equality qℓ = m can be read as [ B : L ][ L : F ] = [ A : F ]. But B ⊆ A and C ⊆ A ⊗ F A op , so B and C are not equal. emma 3.7. Let A be a central simple F -algebra and L a field extension that is strictly finite over F .If L splits A , then we can construct an algebra B ∼ F A such that L is (isomorphic to) a maximal subfieldof B . Moreover [ B : F ] = [ L : F ] .Proof. We follow the proof in Blanchard 1972, Proposition III-4. We let [ L : F ] = ℓ . Since A L ≃ M q ( L ),we have [ A L : L ] = [ A : F ] = q . We have L ⊗ F A ≃ End L ( V ), where V is an L -vector space ofdimension q . We see V as a F -vector space of dimension ℓq . The algebra A ≃ F ⊗ F A ⊆ L ⊗ F A ≃ End L ( V ) is isomorphic to a subalgebra A of D := End F ( V ). By Theorem 2.13, D = A ⊗ F C where C =Z D ( A ). Moreover C is a central simple F -algebra and, since L ⊗ F F and F ⊗ F A commute as subalgebrasof L ⊗ F A , L is isomorphic to a subfield of C . We have ( ℓq ) = [End F ( V ) : F ] = [ A : F ][ C : F ], so[ C : F ] = ℓ = [ L : F ] . So, by Theorem 3.6, L is (isomorphic to) a maximal subfield of C . Finally, since A ⊗ F C ≃ A ⊗ F C ∼ F [ F ] ∼ F C op ⊗ F C, we get A ∼ F C op and C op is the algebra we are looking for. The dynamical method (Mannaa and Coquand 2013; Lombardi and Quitt´e 2015) allows us to work witha commutative algebra K as if it was a discrete field. If, doing so, we find an obstacle in a proof, thismeans that we have found a nonzero noninvertible element a in K . In this case we replace K by K/ h a i ,which is a nonzero quotient that is still a commutative algebra. Moreover, the computations in K thathave been done before the obstacle remain valid in K/ h a i . Splitting central simple algebras
Lemma 4.1. If A is a central simple F -algebra and F is algebraically closed, then A is split.Proof. Our proof is by induction on the F -dimension m of A , the case m = 1 being trivial. If m > f ( X ) ∈ F [ X ] of ε may be written ( X − a ) · · · ( X − a r ) with r > a i ’sin F . So ( ε − a ε ) · · · ( ε − a r ε ) = 0. Since the factors are nonzero, ε − a ε is a nonzero noninvertibleelement of A . By Wedderburn’s theorem 2.4, A ≃ M q ( B ) for some q >
1. We are done by induction.Using Corollary 2.8, we deduce that if F is embedded in an algebraically closed field K and A is acentral simple algebra over F then K splits A .Since the reasoning of the previous sections are all valid intuitionistically, we can interpret this resultdynamically. We obtain in this way the following result.
Theorem and definition 4.2 (central simple algebras are splittable) . Let A be a central simple F -algebra. We can construct a commutative F -algebra K such that A K ≃ M q ( K ) . As an importantcorollary, the F -dimension of A , which is equal to the K -dimension of A K , is always a square: [ A : F ] =[ A K : K ] = q . The integer q is called the degree of the central simple algebra A and is denoted by deg F ( A ) = deg( A ) . A discrete field K is said to be fully factorial when any polynomial in L [ X ] with L any strictly finitefield extension of K can be factorised into irreducible factors (see Mines, Richman, and Ruitenburg 1988,Chapter VII). In the case where F is fully factorial, we obtain the following variation of Theorem 4.2. Theorem 4.3.
Let F be a fully factorial field and A a central simple F -algebra. We can construct afield extension K of F that is strictly finite and splits A .Remark . Following Mannaa and Coquand (2013), the F -algebra K of Theorem 4.2 is obtained in atriangular form F [ x , . . . , x ℓ ] = F [ X , . . . , X ℓ ] / ( P ( X ) , P ( X , X ) , . . . ) with each P i (( x j ) j X i in the polynomial ring F [( x j ) j
We have dynamically an extension K of F such that A K is a matrix algebra M q ( K ). We then write σ ( ε i ) = P k σ ik ε k (with the σ ik ’s in F ) and we extend σ into a K -automorphism σ K of M q ( K ). In fact, σ K is a surjective K -endomorphism. By Proposition1.7, σ K is an inner automorphism given by an u = P i u i ε i with the u i in K .Now we consider the F -linear system of equations σ ( ε i ) v = vε i for i = 1 , . . . , m in the m unknowns v ∈ F m ≃ A . Seeing this system as a K -linear system in the m unknowns v ∈ K m ≃ A K , it hasthe solution given by u ∈ K m . Another solution v satisfies uε i u − v = vε i , so u − v commutes with allelements of A , i.e. u − v ∈ K , or v ∈ Ku . So the solutions of the system over K are the elements of the K -vector space Ku . But since the linear system is a F -system, its solutions in K m are obtained from itssolutions in F m by scalar extension. So the solutions of the system over F are the elements of a F -vectorspace F w ( w ∈ A ) such that Kw = Ku . Moreover xw = 0 implies xu = 0 which implies x = 0, so w isleft regular, hence invertible.We now prove in a similar way that a (dynamical) extension K that splits A can be constructedas a separable F -algebra. This means that K is obtained by successive additions of roots of separablepolynomials. This implies separability of K/F with the meaning that the discriminant of
K/F , i.e. theGram determinant of the trace form ( x, y ) Tr K/F ( xy ), is an invertible element in F (see e.g. Lombardiand Quitt´e 2015, II-5.33 and II-5.36).A discrete field K is said to be separably closed if any separable polynomial is split. In caractersitic p , the only irreducible polynomials over separably closed fields are of the form X p ℓ − d .We follow the argument in Blanchard 1972. Lemma 4.5. If F is separably closed and A is a central simple algebra over F then A is split.Proof. First we assume that A is a division algebra. If A = F there is nothing to do. We will show thatdeg F ( A ) > m = deg F ( A ) >
1, the minimal polynomial f ( X ) of ε is irreducible ofdegree >
1. This polynomial is of the form X p ℓ − d , where p is the characteristic of F and ℓ >
0. So a = ε p ℓ − satisfies a / ∈ F and a p ∈ F . Let σ = Int( a ). We have ( σ − Id A ) = 0 and ( σ − Id A ) p = σ p − Id A = 0.Let b such that ( σ − Id A )( b ) = c = 0 and ( σ − Id A )( c ) = 0. Let u = c − b ; we have σ ( u ) = 1 + u . But u q ∈ F for some q = p n , so σ ( u q ) = 1 + u q since σ is an automorphism of A and σ ( u q ) = u q since u q ∈ F = Z( A ).Let us see the general case, i.e. without assuming that A is a division algebra. Our proof is by inductionon m = deg F ( A ), the case m = 1 being trivial. We redo the preceding proof. If we find an obstacle, thismeans that we have found a nonzero noninvertible element in A , so A ≃ M q ( B ) with a central simplealgebra B and q >
1. We are done by induction.As before, reading dynamically this proof (compare Mannaa and Coquand 2013), we obtain thefollowing result and a variation.
Theorem 4.6 (central simple algebras are splittable by separable commutative algebras) . If A is acentral simple algebra over a field F , we can construct a commutative separable F -algebra K whichsplits F . This extension is obtained in a triangular form F [ x , . . . , x k ] = F [ X , . . . , X k ] / ( P ( X ) , P ( X , X ) , . . . ) with each P i (( x j ) j
Let F be a separably factorial field and A a central simple F -algebra. We can constructa separable field extension K of F that is strictly finite and splits A . This extension is obtained in atriangular form. alois theory In the classical context of a Galois extension L of F generated by the roots of a given separable polyno-mial f , we need to use a dynamical way of computing inside L when there is no algorithm to computeexactly [ L : F ]. A first dynamical approximation of L is given by the universal splitting algebra K of f with Galois group S n ( n = deg( f )). Let us state an elementary result concerning fixed points of theaction of S n . Lemma 4.8 (fixed points under S n in the universal splitting algebra) . Let f be a separable polynomialin F [ X ] and x , . . . , x n the formal roots of f in the universal splitting algebra K of f . Let g ∈ F [ X ] .If g ( x ) = · · · = g ( x n ) in K , then g ( x ) is in F .Proof. Since f ( x i ) = 0 we can take g such that deg( g ) < deg( f ). The polynomial h = g − g ( x ) hasdegree < n and annihilates the n elements x , . . . , x n . Since f is separable, the x i − x j are invertible for i = j , so h = 0 (by Lagrange interpolation), hence g = g ( x ) is in F [ X ] ∩ K = F .NB. In fact, since f is the minimal polynomial of x in K , the proof gives g ( x ) = g (0).The previous lemma is a particular case of a more general result (see Lombardi and Quitt´e 2015,Theorem VII-4.9), but Lemma 4.8 is sufficient for this paper. Reduced trace and reduced norm
In this paragraph we prove the following fact.
Proposition and definition 4.9.
Let A be a central simple F -algebra of degree r . Then the character-istic polynomial of (left multiplication by) an element a of A is a polynomial (of degree n = r ) which isequal to P r ( T ) for a unique monic polynomial P ∈ F [ T ] . Moreover P ( a ) = 0 . • This polynomial P is called the reduced characteristic polynomial of a , and will be denoted by Cprd
A/F ( a )( T ) or Cprd( a )( T ) when the context is clear. • Let P = X r + P rk =1 ( − k s k X r − k . The coefficient s , denoted by Trd( a ) , is called the reducedtrace of a (over F ) . • The coefficient s r , denoted by Nrd( a ) , is called the reduced norm of a (over F ) .Proof. Existence of the reduced characteristic polynomial P and the fact that P ( a ) = 0 are clear when A is split since then P can be taken as the characteristic polynomial of the matrix ϕ ( a ) for an isomorphism ϕ : A → M r ( F ). In this case the reduced trace and the reduced norm are the trace and the determinantof the matrix ϕ ( a ).Uniqueness if existence comes from the fact that F [ T ] is a gcd-domain, and the fact that the divisibilitygroup of a gcd-domain is a lattice-group, which is always torsion-free.We prove now the existence of P ∈ F [ T ] in the general situation.E.g. let us see the case where A is split by a separable commutative algebra L , L = F [ x, y ] = K [ y ] = K [ Y ] / h f ( x, Y ) i , K = F [ x ] = F [ X ] / h f ( X ) i , where f is monic separable over F and f is monic separable over K .Since we have an isomorphism ϕ : A L ∼ −→ M r ( L ), the polynomial Q is equal to P r where P isthe characteristic polynomial of the matrix ϕ ( a ). Moreover P ( ϕ ( a )) = 0, so P ( a ) = 0. We have toprove that P ∈ F [ T ]. We consider the universal splitting algebra B of f over F . In B we have f ( X ) = ( X − x ) · · · ( X − x k ) with x i − x i ′ invertible when i = i ′ . For each i we consider the universalsplitting algebra C i of f ( x i , Y ) over B . In C i we have f ( x i , Y ) = ( Y − y i ) · · · ( Y − y iℓ ) with y ij − y ij ′ invertible when j = j ′ . We note that L = F [ x, y ] ≃ F [ x , y i ] ≃ F [ x i , y ij ].Let us consider a fixed i . Over C i [ T ], we have Q ( T ) = P ( x i , y ij , T ) r for each j . Since C i is re-duced zero-dimensional, the equality of two P ( x i , y ij , T ) r implies the equality of the corresponding It may happen that f has more than n roots in K . We call formal roots the roots x i constructed so as to satisfy theequality f ( X ) = ( X − x ) · · · ( X − x n ) in K [ X ]. Reduced zero-dimensional rings are very similar to discrete fields and share many of their properties (see Lombardiand Quitt´e 2015, Section IV-8 and the elementary local-global machinery no. 2 therein). ( x i , y ij , T ). Now Lemma 4.8 implies that P ( x i , y i , T ) = P ( x i , y ij , T ) has coefficients in F [ x i ] ⊆ B ; wedenote this polynomial by P ( x i , T ). Similarly, since P ( x i , T ) r = Q ( T ) ∈ B [ T ] for each i and B is reduced zero-dimensional, all P ( x i , T )are equal, and Lemma 4.8 implies that these polynomials are in F [ X ]. We denote them by P ( T ) and wehave P r = Q .Such an elementary use of Galois theory can be traced back to Chˆatelet (1946). Lemma 4.10.
Nrd( ab ) = Nrd( a )Nrd( b ) and a is invertible iff Nrd( a ) is nonzero.In this case a − = e a Nrd( a ) − , where e a is equal to R ( a ) with ( − r − P ( T ) = T R ( T ) − Nrd( a ) .Proof. In the previous proof we have ϕ ( ab ) = ϕ ( a ) ϕ ( b ); taking the determinants we get Nrd( ab ) =Nrd( a )Nrd( b ). The last assertion follows from P ( a ) = 0. Involutions of central simple algebras
Reference: Scharlau 1985, Chapter 8.We assume char( K ) = 2.An involution of a K -algebra A is a map J : A → A satisfying J ( a + b ) = J ( a )+ J ( b ), J ( ab ) = J ( b ) J ( a )and J = Id A . We use the notation J ( a ) = a J .We let A + J = A + := { x ∈ A : x J = x } and A − J = A − := { x ∈ A : x J + x = 0 } .We have A + = { ( x + x J ) : x ∈ A } and A − = { ( x − x J ) : x ∈ A } . So A = A + ⊕ A − .If A is central, then J ( K ) = K and F = K ∩ A + is a subfield with F = K or [ K : F ] = 2. If F = K we say that the involution is of the first kind ; if [ K : F ] = 2 we say that the involution is of the secondkind or hermitian . In each case, the product of two involutions of the same kind is a K -automorphism.If A is central simple, this product is an inner automorphism Int( c ).We describe now the usual constructive classification of involutions of the first kind of central simplealgebras. We begin with the case of a split algebra. Lemma 4.11.
Let A = M n ( K ) and J an involution of the first kind (i.e. a K -linear involution). Recallthat A = A + J ⊕ A − J . There exists b ∈ A × such that J = Int( b ) ◦ t and b t = ± b . There are two cases.1. In the first case, the matrix b is symmetric and [ A + J : K ] = n ( n + 1) / ( J is said to be orthogonal ).2. In the second case, the matrix b is skew-symmetric, n is even and [ A + J : K ] = n ( n − / ( J issaid to be symplectic ). Moreover, the characteristic polynomial Cp( a )( T ) of any matrix a ∈ A + J is equal to the square of the so-called Pfaffian characteristic polynomial Cpf( a )( T ) of a : it is theunique monic polynomial in K [ T ] such that Cpf( a ) = Cp( a ) .Proof. When A = M n ( K ), the transposition t : A → A is an involution and J ◦ t = ( t ◦ J ) − is anautomorphism of A , equal to Int( b ) for some b ∈ A × ; in other words J ( a t ) = bab − , or J ( a ) = b a t b − .Writing J = Id A we find x = ( x J ) J = b ( x J ) t b − = b ( bx t b − ) t b − = v − xv with v = b t b − . Since Int( v ) = Id A we have b t b − = λ ∈ K , i.e. b t = λb , which gives b = λb t = λ b .So λ = ± J = Int( b ) ◦ t with b t = ± b . Note that b is the matrix of some nondegenerate bilinearform on K n , ( x, y ) x t by , which is symmetric or skew-symmetric.If b is symmetric ( J is orthogonal), we get A + J = { a ∈ A : ( ab ) t = ab } , so [ A + J : K ] = n ( n + 1) / b is skew-symmetric ( J is symplectic), n has to be even. We get A + J = { a ∈ A : ( ab ) t = − ab } , so[ A + J : K ] = n ( n − /
2. Knus, Merkurjev, Rost, and Tignol (1998, Proposition (2.9)) provide the rest ofthe lemma.In the two cases, for a given J , the matrix b and the nondegenerate (symmetric or skew-symmetric)bilinear form defined by b are well-defined up to a scalar multiplicative factor.Now we state the general theorem. Lemma 4.8 uses a discrete field F . Since F [ x i ] ≃ F [ x ] may contain nonzero noninvertible elements, we have perhapsto replace f by a proper factor. Here x and y are viewed as column vectors. roposition 4.12. Let A be a central simple K -algebra of degree n > and J an involution of the firstkind (i.e. a K -linear involution). Recall that A = A + J ⊕ A − J . There are two cases.1. In the first case, [ A + J : K ] = n ( n + 1) / ( J is said to be orthogonal).2. In the second case, n is even and [ A + J : K ] = n ( n − / ( J is said to be symplectic). Moreover thereduced characteristic polynomial Cprd( a )( T ) of any a ∈ A + J is equal to the square of the so-called Pfaffian characteristic polynomial Cpf( a )( T ) of a ; Cpf( a )( T ) is the unique monic polynomial in K [ T ] such that Cpf( a ) = Cprd( a ) .Proof. In the general case the central simple K -algebra A is split by a commutative K -algebra L .Assume first that L is a field. Then the proposition is satisfied for A L and the extension J L of J to A L . These assertions transfer clearly to J .In the general case, we cannot force constructively L to be a field, but dynamically we obtain theresult as if L was a field: when an obstruction in the proof appears with a nonzero noninvertible element z ∈ L , we replace L by L/ h z i . We denote by Br ( F ) the subgroup of Br( F ) of elements [ A ] or order 2 (i.e. such that [ A ] = 1).In this final section, F is a discrete field of characteristic = 2. We will prove two constructive versionsof the following result in Becher 2016, Theorem. Theorem 5.1.
Let F be a discrete field of characteristic = 2 . Every element of Br ( F ) is split by afinite extension of F obtained by a tower of quadratic extensions. Some results about quaternion algebras
When F has characteristic = 2, a quaternion algebra is defined as the 4-dimensional F -algebra h F ( a, b ) =h( a, b ) with basis (1 , α, β, γ ), the multiplication table being determined by α = a, β = b, γ = αβ = − βα where a, b ∈ F × . This is a central simple algebra.We have αγ = − γα , βγ = − γβ and γ = − ab . It is generated as a F -algebra by α and β . If q = x + yα + zβ + wγ ∈ h( a, b ) we let q I = x − yα − zβ − wγ and N( q ) = qq I = x − ay − bz + abw ∈ F. The map q q I is an involution of h( a, b ) and N( q q ) = N( q )N( q ). Moreover q is invertible if, andonly if, N( q ) = 0. A pure quaternion is a quaternion such that x = 0. It is characterised by q / ∈ F whereas q ∈ F (or q = 0), or also by q I = − q , or also by N( q ) = − q .The following theorem recalls basic results which are given with a constructive proof in most textbooks(e.g. Gille and Szamuely 2006, Chapter 1). Theorem 5.2 (basics about quaternion algebras) . h( a, b ) ≃ h( u a, v b ) ( u, v ∈ F × ) .2. h( a, b ) ≃ h( b, a ) ≃ h( a, − ab ) ≃ h( b, − ab ) .3. h(1 , b ) is split (isomorphic to M ( F ) ), so h( u , b ) is split.4. h( a, − a ) ≃ h(1 , if a = 0 , .5. h( a, b ) ⊗ F h( a ′ , b ) ≃ h( aa ′ , b ) ⊗ F M ( F ) .6. h( a, b ) ⊗ F h( a, b ′ ) ≃ h( a, bb ′ ) ⊗ F M ( F ) .7. The following are equivalent. • h( a, b ) is split. • N : h( a, b ) → F has a nontrivial zero. • the conic ax + by − z = 0 has a F -rational point (in P ( F ) ).8. The following are equivalent. • h( a, b ) is a division algebra. N : h( a, b ) → F has no nontrivial zero. • the conic ax + by − z = 0 has no F -rational point (in P ( F ) ). • h( a, b ) is not split.9. If a is not a square in F , the following are equivalent. • h( a, b ) is split. • ∃ c, d ∈ F, a = c − d b . Lemma 5.3.
A central F -algebra of dimension is a quaternion algebra ( char( F ) = 2 ).Proof. Let z ∈ A \ F ; F [ z ] is a commutative field of F -dimension d = 2 since d divides 4 and is neither 1nor 4. Since char( F ) = 2 we have an x such that F [ z ] = F [ x ] and x = a ∈ F . The linear map ρ x : v x − vx is an automorphism of order ≤
2. But ρ x = Id A since xv = vx for some v . So there is aneigenvector y such that ρ x ( y ) = − y and we get a F -basis (1 , x, y, xy ). Then y x = y ( − xy ) = − ( yx ) y = xy , so y = b is in the center F . We conclude that A ≃ h( a, b ).In the following, except for the second constructive version of Becher’s theorem, we assume that F isa fully factorial field, or even, since this has the same consequences for algorithms, that F is containedin an algebraically closed discrete field L such that F is detachable in L . First step
This step corresponds to Becher 2016, Lemma 1 (Lemma 5.8 below), and the proof will follow its mainidea.
Context 5.4.
In this first step we consider a quadratic field extension K of F . Since char( F ) = 2 wecan write K = F [ δ ] with δ = g ∈ F and δ / ∈ F . See Scharlau 1985, Section 8.9.We have the F -automorphism σ : x + yδ x − yδ of K ( x, y ∈ F ). We use the notation ¯ w = σ ( w ).We consider an arbitrary quaternion K -algebra A = h K ( a + bδ, c + dδ ) with a, b, c, d ∈ F , ( a, b ) = (0 , c, d ) = (0 , a = b g and c = d g (since δ / ∈ F ). We consider the dual quaternion K -algebra ¯ A = h K ( a − bδ, c − dδ ). The generators of A and ¯ A satisfy the following equalities. u = a + bδ v = c + dδ uv = − vu. ¯ u = a − bδ ¯ v = c − dδ ¯ u ¯ v = − ¯ v ¯ u. The K -semilinear map G : w ¯ w, A → ¯ A defined on the K -basis (1 A , u, v, uv ) by1 A ¯ A , u ¯ u, v ¯ v and uv ¯ u ¯ v is a F -isomorphism from A to ¯ A , i.e. when we identify K with its image in A , it is an isomorphism of F -algebras and zw = ¯ z ¯ w if z ∈ K and w ∈ A .Let us consider the central simple K -algebra B := A ⊗ K ¯ A . The automorphism I B of the F -vectorspace B defined by e ⊗ f G − ( f ) ⊗ G ( e ) is a ring automorphism of order 2 which satisfies I B ( z ) = ¯ z if z ∈ K .Since I B = Id B , we have B = Ker( I B − Id B ) ⊕ Ker( I B + Id B ). Moreover, since I B ( δ ) = − δ , the K -linear map w δw permutes the two summands. So the invariant elements form a sub- F -vectorspace of dimension 16: T = Ker( I B − Id B ) is a sub- F -algebra such that B = T ⊕ δT = K ⊗ F T . ByCorollary 2.8, T is a central simple algebra of degree 4 over F ; it is called the corestriction of A over F . Lemma 5.5. T contains a quaternion subalgebra, and hence is the tensor product of two quaternion F -algebras.Proof. We take x = v ¯ v and y = ( u + ¯ u )( cb − ad + du ¯ u + bv ¯ v ) and we verify directly that x and y belongto F and xy = − yx . The consequence is obtained by applying Theorem 2.13 and Lemma 5.3. Definition 5.6 (Lam) . We say that a field F satisfies Property A if every quaternion F -algebra is split. Lemma 5.7. If F satisfies Property A and A ⊗ K ¯ A is split, then A is the extension of a quaternion F -algebra and is therefore split. roof. Since A ⊗ K ¯ A is split, it is isomorphic to A ⊗ K A op (Theorem 2.12) and we get a K -algebraisomorphism J : A → ¯ A op . Composing J with the F -algebra isomorphism ¯ A → A we get a F -linearisomorphism H : A → A op satisfying H ( z ) = ¯ z if z ∈ K and H ( xy ) = H ( y ) H ( x ) for all x, y ∈ A . Considerthe canonical K -linear involution I : A → A defined by I ( u ) = − u, I ( v ) = − v, I ( uv ) = − uv = I ( v ) I ( u ).It is clear that C := Ker( H − I ) is a sub- F -algebra of A and δC ⊆ Ker( H + I ).Assume HI = IH ; then C ⊕ δC = A since in this case 0 = ( H + I )( H − I ) and hence A = Ker( H − I ) ⊕ Ker( H + I ). We know that C is a quaternion F -algebra by Lemma 5.3, so it is split over F byhypothesis. This implies that A = K ⊗ F C is split over K .It remains to prove HI = IH . For x in A \ K , the element I ( x ) is the unique element in A suchthat both x + I ( x ) and xI ( x ) = I ( x ) x are in K . Then H ( x + I ( x )) = H ( x ) + H ( I ( x )) ∈ K and H ( x ) H ( I ( x )) = H ( I ( x ) x ) = H ( xI ( x )) = H ( I ( x )) H ( x ) ∈ K , and so H ( I ( x )) = I ( H ( x )). For x in K wehave H ( I ( x )) = σ ( x ) = I ( H ( x )). Lemma 5.8. If F satisfies Property A then so does K . Note also that this can be interpreted as follows: if any conic over F has a rational point, then thesame holds for K . Proof.
We consider a quaternion algebra A over K . Using Lemma 5.7 it is enough to show that B = A ⊗ K ¯ A is split. Lemma 5.5 says that T is the tensor product of two quaternion F -algebras, so T is split,and the result follows from the fact that B = K ⊗ F T . Corollary 5.9. If F satisfies Property A and a field K obtained from F by a sequence of quadraticextensions splits A , then A is already split over F .Proof. Using Lemma 5.8 we are reduced to the case where K is a quadratic extension of F . By Lemma 3.7we find B ∼ F A such that K ⊆ B and [ B : F ] = 4. By Lemma 5.3, B is a quaternion algebra and it issplit by Property A . So A is split.The proof of Corollary 5.9 is constructive and can be seen as an algorithm. So there are only a finitenumber of quaternion algebras split by F that are used in the algorithm. Precisely, we have the followingalgorithmic consequence of Corollary 5.9. Corollary 5.10.
Assume we have a field extension K of F such that • K = F [ x , . . . , x n ] with each x i ∈ F [( x j ) j
This corresponds to Becher 2016, Proposition, and the proof can be taken as it stands.
Lemma 5.11.
Let K be a discrete field extension of F with a primitive element t of degree N > .Assume that any polynomial of degree < N has a root in F ; then K satisfies Property A .Proof. Note that the hypothesis implies in particular that any element of F is a square. We write K = F [ t ]. Any element of K can be written in the form f ( t ) with f of degree < N , and hence can bewritten as f ( t ) = c ( t − a ) · · · ( t − a k ) with c a square. Moreover, any quaternion algebra h K ( t − a, t − b )is split since b − a is a square: the conic ( t − a ) x + ( t − b ) y − z has a nontrivial zero (1 , ι, u ) where The commutative K -algebra K = K [ x ] is a quadratic field extension of K and the conjugate of x for the corresponding K -automorphism of K is the unique element y of K satisfying x + y ∈ K and xy = yx ∈ K . In fact, K = Z A ( K ), so y is also the unique element of A satisfying these properties. Finally, a direct computation shows that x + I ( x ) ∈ K and xI ( x ) = I ( x ) x ∈ K . = − u = b − a . It follows that in general h K ( f ( t ) , g ( t )) is split: if g ( t ) = d ( t − b ) · · · ( t − b ℓ ),writing [ u, v ] for [h K ( u, v )] Br we get [ f ( t ) , g ( t )] = Y ℓj =1 [ c, ( t − b j )] Y ki =1 [( t − a i ) , d ] Y ki =1 Y ℓj =1 [ t − a i , t − b j ]and all the terms in the products vanish.Now we give an algorithmic consequence which relates to Lemma 5.11 as Corollary 5.10 does toCorollary 5.9. Corollary 5.12.
Let K be a discrete field extension of F with a primitive element t of degree N > .In order to split a given finite number of quaternion algebras over K , it suffices to extend F by addinga finite number of roots of polynomials of degree < N . Third step
This step corresponds to Becher 2016, Lemma 4, but the proof is different.
Lemma 5.13.
Assume that every element of F is a square. If A ∈ Br ( F ) then A is split or it has asymplectic involution. NB. In the last case, deg( A ) is even. So, if deg( A ) is odd, A is split. Proof.
Let N := deg( A ). We use an induction on N , the case N = 1 being trivial. In the case N = 2we have a quaternion algebra (Lemma 5.3) and it is split (Theorem 5.2 item ). Now we assume N > A ] = 1 in Br( F ) means that A ⊗ F A is split (Theorem 3.3 item ), hence isomorphicto A ⊗ F A op and we have an isomorphism J : A → A op (Theorem 3.3 item ). So J is an automorphismof the F -algebra A and hence, by the Skolem-Noether theorem, there is an element α in A × such that J = Int( α − ). We have for all x ∈ Aα − J ( x ) α = J ( J ( x )) = J ( x ) = J ( J ( x )) = J ( α − xα ) = J ( α ) J ( x ) J ( α ) − , so t = αJ ( α ) ∈ Z F ( A ) = F . If α ∈ F we have finished: J is an involution. Now, if α / ∈ F , since t = a with a ∈ F × , we replace α with a − α and get αJ ( α ) = 1. We seek β ∈ A × such that Int( β ) ◦ J is aninvolution:(Int( β ) ◦ J ) ( x ) = βJ (cid:0) βJ ( x ) β − (cid:1) β − = βJ ( β − ) J ( x ) J ( β ) β − = βJ ( β − ) α − xαJ ( β ) β − = γ − xγ. So we need that γ = αJ ( β ) β − ∈ F . Let β = 1 + α : β is nonzero since α / ∈ F . If β is noninvertible, A = M q ( B ) for some B and q > β is invertible, we are done since αJ (1 + α ) = α (1 + J ( α )) = α + 1 = β .If the involution J ′ we have found is symplectic we have finished. If J ′ is orthogonal, since [ A − J ′ : F ] = N ( N − / ≥
3, there exists a nonzero y ∈ A such that y + J ′ ( y ) = 0. If y is noninvertible, we aredone by induction. If y is invertible, I = Int( y ) ◦ J ′ is a symplectic involution. In fact, I is clearly aninvolution and a = I ( a ) ⇐⇒ ay = yJ ′ ( a ) ⇐⇒ ay = − J ′ ( y ) J ′ ( a ) ⇐⇒ ay = − J ′ ( ay ) ⇐⇒ ay ∈ A − J ′ , so that [ A − J ′ : F ] = [ A + I : F ] = N ( N − / Lemma 5.14.
Let
N > and assume that every monic polynomial in F [ X ] of degree < N has a rootin F . Then any central simple F -algebra B of degree < N is split.Proof. Same proof as in Lemma 4.1.
Lemma 5.15.
Let
N > and assume that every polynomial of degree ≤ sup(2 , N/ has a root in F .Let K be an extension of degree N of F . Let A ∈ Br ( F ) such that A is split by K . Then A is alreadysplit by F . Note that Theorem 5.2 shows that [ uu ′ , v ] = [ u, v ][ u ′ , v ] and [ u, vv ′ ] = [ u, v ][ u, v ′ ]. roof. We can find an algebra B ∼ F A which contains K as a maximal subfield with deg F ( B ) = N byLemma 3.7. We have to show that B is split over F . Using Lemma 5.13, either B is split, or N is even andwe find a symplectic involution J on B . We have [ B + J : F ] = N ( N − / ≥
6, and any x ∈ B + J \ F satisfiesCpf( x )( x ) = 0 with deg(Cpf( x )) = N/
2. As in Lemma 4.1 we find a nonzero noninvertible element in B .So B = M r ( B ) with r >
1, and Lemma 5.14 applies for B since deg F ( B ) ≤ N/ < N .Now we give an algorithmic consequence of Lemma 5.15 in line with Corollaries 5.10 and 5.12. Corollary 5.16.
Let K be an extension of degree N of F . Let A ∈ Br ( F ) such that A is split by K .Then A is already split by F if we add to F successively a finite number of roots of polynomials of degree ≤ sup(2 , N/ . Constructive versions of Theorem 5.1
Our first constructive version is with the hypothesis made for section 5: the discrete field F is supposedto be fully factorial. Theorem 5.17.
Let F be a fully factorial discrete field of characteristic = 2 . Every element of Br ( F ) is split by a strictly finite extension of F obtained by a tower of quadratic field extensions.Proof. We can combine the results of the three previous subsections to reduce a splitting sequence to aquadratic splitting sequence.The argument will be the following: we start with a field L = F [ x , . . . , x ℓ ] which splits A , givenexplicitly by Theorem 4.2. Each x i is a root of a monic polynomial of degree n i over F [( x j ) j
2) with
N > σ, m , . . . , m r )where all m j are < N . This means that the new sequence has strictly decreased for an order type ω ω given by ( p , . . . , p r ) > ( q , . . . , q s ) if the multiset defined by ( p , . . . , p r ) is strictly greater than themultiset defined by ( q , . . . , q s ) for the lexicographic order. So our algorithm stops in a finite numberof steps.Now we explain how we manage to replace ( σ, N, , . . . ,
2) by ( σ, m , . . . , m r ).Let K be the field constructed with the sequence ( σ ) and K = K [ y ] the field constructed withthe sequence ( σ, N ). The field K splits A after a sequence of quadratic extensions. In order that K splits A , it is sufficient, by Corollary 5.10, that it splits certain (finitely many) quaternion algebras.By Corollary 5.12 it suffices to extend K into K ′ by adding some roots of polynomials of degree < N in order that the corresponding extension K ′ = K ′ [ y ] of K splits the given quaternion algebras. In thiscase, K ′ splits A by Corollary 5.10. By Corollary 5.16, it suffices to extend K ′ into K ′′ by adding someroots of polynomials of degree ≤ sup(2 , N/
2) in order that K ′′ splits A .In the second constructive version we don’t assume that F is fully factorial. The conclusion uses nowa tower of quadratic algebra extensions. Theorem 5.18.
Let F be a discrete field of characteristic = 2 . Every element of Br ( F ) is split by acommutative F -algebra K obtained by a tower of quadratic algebra extensions.Proof. This follows from the proof of Theorem 5.17, by using the dynamical way of transforming thealgorithms.Note that if, after the end of our algorithm/construction, we find a nonzero noninvertible elementin K , we are able to simplify the tower by cancelling one or several (useless) extensions K i +1 ≃ K i × K i .The improvement of Theorem 5.18 w.r.t. Theorem 5.17 is that there is no need to factorise polynomialsin order to get the result, and the new version is nevertheless directly equivalent to Theorem 5.1 inclassical mathematics. This means that we replace F by F [ y , . . . , y ℓ ] with each y j of degree ≤ sup(2 , N/
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