Proof: This follows from the theorem in view of the calculation of the cohomology of cyclic groups (Example 3.2.9).
Finally, we also record the following inflation-restriction sequence:
Corollary 4.4.11 For a finite Galois extension K|k there is an exact se-quence
0→Br (K|k)−→Inf Br (k)−→Res Br (K).
Proof: This follows from the theorem and Corollary 4.3.5 in degree 2 (which applies to the Gal (ks|k)-module ks× in view of Hilbert’s Theorem 90).
4.5 Index and Period 119 Proof: Let D◦ be the opposite algebra to D. We have established during the proof of Proposition 2.4.8 an isomorphism D⊗k D◦ ∼= Endk(D). If K is as above, the inclusion K ⊂ D induces an inclusion K ⊂ D◦ by commu-tativity of K, whence also an injection ι : D ⊗k K → Endk(D). As the endomorphisms of D coming from D ⊗kK are given by multiplication by elements of K, we see that the image ofιlies in EndK(D). By definition, we have EndK(D)∼=Mn(K), wheren = indk(D); in particular, it has dimension n2 over K. On the other hand, we have dimK(D⊗kK) = dimk(D) = n2, so the map ι: D⊗kK →EndK(D) is an isomorphism.
We can now prove the following basic fact.
Proposition 4.5.4 Every central simple k-algebra A is split by a separable extension K|k of degree ind(A) over k. Moreover, such a K may be found among the k-subalgebras of A.
The proof is based on the following lemma, which uses the notion of the reduced characteristic polynomial Pa(T) of an element a ∈ A. This is defined as the polynomial Nrd(T−a)∈k[T], where Nrd is the reduced norm map introduced in Construction 2.6.1. Note that if we choose an algebraic closure ¯k ofk and an isomorphismA⊗k¯k∼=Mn(¯k), then Pa(T) becomes the characteristic polynomial of the matrixMacorresponding toa. In particular, its coefficients are polynomials in the entries of Ma.
Lemma 4.5.5 ForA as above, we may find a∈A so that its reduced char-acteristic polynomialPa(T) has distinct roots.
Proof: The polynomialPa(T) has distinct roots if and only if its discrimi-nantDa is nonzero. It is known from algebra thatDa is a polynomial in the coefficients ofPa(T). Now choose an isomorphism A⊗k¯k∼=Mn(¯k) and view the elements Mn(¯k) as points of affine n2-space over ¯k. By the above discus-sion, the points corresponding to matrices whose characteristic polynomial has nonzero discriminant form a Zariski open subset in An¯k2. A k-rational point in this open subset corresponds to an element a∈A with the required property.
Proof of Proposition 4.5.4: By Wedderburn’s theorem we may assume that A is a division algebra. By the lemma we find a ∈ A so that Pa(T) has distinct roots. As Pa(T) is the characteristic polynomial of a matrixMa over ¯k, this implies thatMa has distinct eigenvalues, and hence Pa(T) is also its minimal polynomial. In particular,Pa(T) is irreducible over ¯k and hence also over k, so the ring K := k[T]/(Pa(T)) is a separable field extension
of k. Therefore the homomorphism k[T] → A sending T to a embeds K as a subfield in A which is of degree degPa(T) = degk(A) = indk(A) over k (as A is assumed to be a division algebra). We conclude by the previous proposition.
To proceed further, we need the following refinement of Theorem 4.4.5.
Proposition 4.5.6 LetK|k be a separable field extension of degreen. Then the boundary map δn: H1(k,PGLn(k))→Br (k) induces a bijection
ker(H1(k,PGLn(k))→H1(K,PGLn(K))) →∼ Br (K|k).
The proof uses a lemma from Galois theory.
Lemma 4.5.7 Let Ke be the Galois closure of K, and denote the Galois groupsGal (Ke|k)andGal (Ke|K)byGandH, respectively. Making Gact on the tensor product K⊗kKe via the second factor, we have an isomorphism of G–modules
(K ⊗kK)e × ∼=MHG(Ke×).
Proof: According to the theorem of the primitive element, we may write K =k(α) for some α ∈ K with minimal polynomial f ∈k[x], so that Ke is the splitting field off. By Galois theory, if 1 =σ1, . . . , σn is a system of left coset representatives for H in G, the roots of f in K are exactly the σi(α) for 1 ≤ i ≤ n. So we get, just like before the proof of Speiser’s lemma in Chapter 2, a chain of isomorphisms
K⊗kKe ∼=K[x]/e Yn i=1
(x−σi(α))∼= HomH(Z[G],Ke) =MHG(Ke).
The lemma follows by restricting to invertible elements.
Proof of Proposition 4.5.6: We have already shown in the proof of The-orem 4.4.5 the injectivity of δn (even ofδ∞), so it suffices to see surjectivity.
With the notations of the lemma above, consider the short exact sequence of G-modules
1→Ke× →(K⊗kK)e ×→(K⊗kK)e ×/Ke× →1,
where G acts on K⊗kKe via the second factor. Part of the associated long exact sequence reads
H1(G,(K⊗kK)e ×/Ke×)→H2(G,Ke×)→H2(G,(K⊗kK)e ×). (5)
4.5 Index and Period 121 Using the previous lemma, Shapiro’s lemma and Theorem 4.4.5, we get a chain of isomorphisms
H2(G,(K⊗kK)e ×)∼=H2(G, MHG(K))e ∼=H2(H,Ke)∼= Br (Ke|K).
We also haveH2(G,Ke×)∼= Br (Ke|k), so all in all we get from exact sequence (5) a surjection
e
α: H1(G,(K⊗kK)e ×/Ke×)→Br (K|k)
On the other hand, the choice of a k-basis of K provides an embedding K ,→Mn(k), whence aG-equivariant map K⊗kKe →Mn(K), and finally ae map (K ⊗kK)e × →GLn(K). Arguing as in the proof of Theorem 4.4.5, wee get a commutative diagram:
H1(G,(K⊗kK)e ×/K×) −−−→αe H2(G,Ke×)
y
yid H1(G,PGLn(Ke)) −−−→δn H2(G,Ke×)
Therefore by the surjectivity of αe each element of Br (K|k) ⊂ H2(G,Ke×) comes from some element in H1(G,PGLn(Ke)). By the injectivity of δn and its obvious compatibility with restriction maps, this element restricts to 1 in H1(H,PGLn(K)), as required.e
We can now prove the following characterisations of the index.
Proposition 4.5.8 LetA be a central simplek-algebra. The indexind(A)is the greatest common divisor of the degrees of finite separable field extensions K|k that split A.
Proof: In view of Proposition 4.5.4 it is enough to show that if a finite sep-arable extensionK|k of degree n splits A, then ind(A) divides n. For such a K, the class of A in Br (K|k) comes from a class inH1(k,PGLn(k)) accord-ing to Proposition 4.5.6. By Theorem 2.4.3 this class is also represented by some central simple k-algebraB of degree n, hence of index dividing n. But ind(A) = ind(B) by Remark 4.5.2 (2).
Combining with Proposition 4.5.4 we get:
Corollary 4.5.9 The indexind(A)is the smallest among the degrees of finite separable field extensions K|k that split A.
Here are some other easy corollaries.
Corollary 4.5.10 LetA andB be central simplek-algebra that generate the same subgroup in Br (k). Then ind(A) = ind(B).
Proof: The proposition implies that for all i we have ind(A⊗i) | ind(A).
But for suitable i and j we have [A⊗i] = [B] and [B⊗j] = [A] in Br (k) by assumption, so the result follows, taking Remark 4.5.2 (2) into account.
Corollary 4.5.11 Let K|k be a finite separable field extension.
1. We have the divisibility relations
indK(A⊗kK)|indk(A)|[K :k] indK(A⊗kK).
2. If indk(A) is prime to [K : k], then indk(A) = indK(A ⊗k K). In particular, if A is a division algebra, then so is A⊗kK.
Proof: It is enough to prove the first statement. The divisibility relation indK(A⊗kK)| indk(A) is immediate from the proposition. For the second one, use Proposition 4.5.4 to find a finite separable field extensionK0|K split-tingA⊗kK with [K0 :K] = indK(A⊗kK). Then K0 is also a splitting field of A, so Proposition 4.5.8 shows indk(A)|[K0 :k] = indK(A⊗kK)[K :k].
Now we come to the second main invariant.
Definition 4.5.12 The period(orexponent) of a central simplek-algebra A is the order of its class in Br (k). We denote it by per(A).
The basic relations between the period and the index are the following.
Proposition 4.5.13 (Brauer) Let A be a central simple k-algebra.
1. The period per(A) divides the index ind(A).
2. The period per(A) and the index ind(A) have the same prime factors.
For the proof of the second statement we shall need the following lemma.
Lemma 4.5.14 Let p be a prime number not dividing per(A). Then A is split by a finite separable extension K|k of degree prime to p.
4.5 Index and Period 123 Proof: LetL|kbe a finite Galois extension that splitsA, letP be ap-Sylow subroup of Gal (L|k) and K its fixed field. Then Br (L|K)∼=H2(P, L×) is a p-primary torsion group by Corollary 4.4.8, so the assumption implies that the image of [A] by the restriction map Br (L|k)→Br (L|K) is trivial. This means that A is split by K.
Proof of Proposition 4.5.13: According to Proposition 4.5.4, the algebra A is split by a separable extension K|k of degree ind(A) over A. By Propo-sition 4.5.6, the class [A] of Ain Br (k) is then annihilated by the restriction map Br (k) → Br (K). Composing with the corestriction Br (k) → Br (K) and using Proposition 4.2.10, we get that [A] is annihilated by multiplication by [K :k] = ind(A), whence the first statement. For the second statement, let p be a prime number that does not divide per(A). By the lemma above, there exists a finite separable splitting field K|k with [K : k] prime to p.
Hence by Proposition 4.5.8, the index ind(A) is also prime top.
Remark 4.5.15 It is an interesting and largely open question to determine the possible values of the integer ind(A)/per(A) for central simple algebras over a given field k. For instance, it is conjectured by Michael Artin [1]
that for C2-fields (see Remark 6.2.2 for this notion) one should always have per(A) = ind(A). The conjecture is now known to hold for arithmetic fields (see Corollary 6.3.10 as well as Remarks 6.5.5 and 6.5.6), function fields of complex surfaces (de Jong [1]), and completions of the latter at smooth points (Colliot-Th´el`ene/Ojanguren/Parimala [1]). Another interesting recent result on this topic is that of Saltman [4], who proves that for an algebra A over the function field of a curve over ap-adic field Qp the ratio ind(A)/per(A) is always at most 2, provided that per(A) is prime to p.
As an application of the above, we finally prove the following decompo-sition result.
Proposition 4.5.16 (Brauer) Let D be a central division algebra over k.
Consider the primary decomposition
ind(D) =pm11pm22· · ·pmrr.
Then we may find central division algebras Di (i= 1, .., r) such that D∼=D1⊗kD2 ⊗k· · · ⊗kDr
and ind(Di) =pmi i for i = 1, .., r. Moreover, the Di are uniquely determined up to isomorphism.
Proof: The Brauer group is torsion (Corollary 4.4.8), so it splits into p-primary components:
Br (k) =M
p
Br (k){p}.
In this decomposition the class of D decomposes as a sum [D] = [D1] + [D2] +· · ·+ [Dr]
where theDi are division algebras with [Di]∈Br (k){pi}for some primes pi. By Proposition 4.5.13 (2) the index of each Di is a power ofpi. The tensor product A = D1⊗k D2 ⊗k · · · ⊗k Dr has degree Q
iind(Di) over k and its index equals that of D by Remark 4.5.2 (2), so indD divides Q
iind(Di). A repeated application of Proposition 4.5.4 shows that for fixedi one may find a finite separable extension Ki|k of degree prime to pi that splits all the Dj
for j 6=i. Then D⊗kKi and Di⊗kKi have the same class in Br (Ki), and thus indKi(Di⊗kKi)|ind(D) by Corollary 4.5.11 (1). The algebrasDi⊗kKi
are still division algebras of index ind(Di) overKi by Corollary 4.5.11 (2). To sum up, we have proven that ind(Di) divides ind(D) for alli, so we conclude that ind(D) = Q
iind(Di). The k-algebrasDandD1⊗kD2⊗k· · ·⊗kDr thus have the same Brauer class and same dimension, hence they are isomorphic as claimed. The unicity of theDi holds for the same reason.