Splitting Fields

The last corollary enables one to give an alternative characterisation of cen-tral simple algebras.

Theorem 2.2.1 Let k be a field and Aa finite dimensionalk-algebra. Then A is a central simple algebra if and only if there exist an integer n > 0 and a finite field extensionK|k so that A⊗kK is isomorphic to the matrix ring Mn(K).

2.2 Splitting Fields 33 We first prove:

Lemma 2.2.2 LetAbe a finite dimensionalk-algebra, andK|k a finite field extension. The algebra A is central simple over k if and only if A⊗^kK is central simple over K.

Proof: IfI is a nontrivial (two-sided) ideal ofA, thenI⊗^kK is a nontrivial ideal of A⊗kK (e.g. for dimension reasons); similarly, if A is not central, then neither is A⊗kK. Thus if A⊗kK is central simple, then so is A.

Using Wedderburn’s theorem, for the converse it will be enough to con-sider the case when A = D is a division algebra. Under this assumption, if w₁, . . . , w_nis ak-basis ofK, then 1⊗w₁, . . . ,1⊗w_nyields aD-basis ofD⊗kK as a leftD-vector space. Given an element x=P

αi(1⊗wi) in the center of D⊗kK, for alld∈Dthe relationx= (d⁻¹⊗1)x(d⊗1) =P

(d⁻¹αid)(1⊗wi) implies d⁻¹α_id=α_i by the linear independence of the 1⊗w_i. As Dis central over k, the αi must lie in k, so D⊗^kK is central over K. Now if J is a nonzero ideal in D⊗kK generated by elements z1, . . . , zr, we may assume thez_i to beD-linearly independent and extend them to aD-basis of D⊗kK by adjoining some of the 1⊗wi, say 1⊗wr+1, . . . ,1⊗wn. Thus for 1≤i≤r we may write

1⊗wi = Xn j=r+1

αij(1⊗wj) +yi,

where y_i is some D-linear combination of the z_i and hence an element of J. Herey1, . . . , yrareD-linearly independent (because so are 1⊗w1, . . . ,1⊗wr), so they form aD-basis ofJ. AsJ is a two-sided ideal, for alld∈Dwe must have d⁻¹y_id ∈J for 1≤ i≤r, so there exist β_il ∈D with d⁻¹y_id =P

β_ily_l. We may rewrite this relation as

(1⊗w_i)− Xn j=r+1

(d⁻¹α_ijd)(1⊗w_j) = Xr

l=1

β_il(1⊗w_l)− Xr

l=1

β_il Xn j=r+1

α_lj(1⊗w_j), from which we get as above, using the independence of the 1⊗w_j, that βii= 1, βil = 0 for l6=i and d⁻¹αijd=αij, i.e. αij ∈k asD is central. This means that J can be generated by elements of K (viewed as a k-subalgebra of D⊗k K via the embedding w 7→ 1⊗w). As K is a field, we must have J∩K =K, so J =D⊗kK. This shows that D⊗kK is simple.

Proof of Theorem 2.2.1: Sufficiency follows from the above lemma and Example 2.1.2. For necessity, note first that denoting by ¯k an algebraic closure of k, the lemma together with Corollary 2.1.7 imply that A⊗k¯k ∼= Mn(¯k) for some n. Now observe that for every finite field extension K of k

contained in ¯k, the inclusionK ⊂k¯induces an injective mapA⊗kK →A⊗k¯k and A ⊗k k¯ arises as the union of the A ⊗k K in this way. Hence for a sufficiently large finite extension K|k contained in ¯k the algebra A ⊗^k K contains the elements e1, . . . , en² ∈ A ⊗k ¯k corresponding to the standard basis elements ofM_n(¯k) via the isomorphismA⊗k¯k∼=M_n(¯k), and moreover the elementsaij occuring in the relationseiej =P

aijei defining the product operation are also contained in K. Mapping the ei to the standard basis elements ofM_n(K) then induces a K-isomorphism A⊗kK ∼=M_n(K).

Corollary 2.2.3 If A is a central simple k-algebra, its dimension over k is a square.

Definition 2.2.4 A field extension K|k over whichA⊗kK is isomorphic to Mn(K) for suitable n is called a splitting field for A. We shall also employ the terminology A splits overK orK splits A.

The integer √

dimkA is called the degreeof A.

The following proposition, though immediate in the case of a perfect base field, is crucial for our considerations to come.

Proposition 2.2.5 (Noether, K¨othe) A central simple k-algebra has a splitting field separable over k.

Proof: Assume there exists a central simple k-algebra A which does not split over any finite separable extension K|k. Fix separable and algebraic closures k^s ⊂ ¯k of k. By the same argument as at the end of the proof of Theorem 2.2.1, the k^s-algebra A ⊗kk^s does not split over k^s, hence by Wedderburn’s theorem it is isomorphic to some matrix algebraM_n(D), where Dis a division algebra over k^s different fromk^s. Letd >1 be the dimension ofD over k^s. Then by Corollary 2.1.7 we have D⊗k^sk¯∼=Md(¯k). Regarding the elements of M_d(¯k) as ¯k-points of affine d²-space A^d2, elements of D correspond to the points ofA^d2 defined overk^s. AsDis a division algebra, its nonzero elements give rise to invertible matrices inMd(¯k); in particular, they have nonzero determinant. Now the map which sends an element of M_d(¯k) viewed as a point ofA^d2(¯k) to its determinant is given by a polynomialP in the variablesx1, . . . , xd²; note thatP ∈k^s[x1, . . . , xd²] as its coefficients are all 1 or−1. Hence our assumption means that the hypersurfaceH ⊂A^d2 defined by the vanishing ofP contains no points defined overk^sexcept for the origin.

But this contradicts the basic fact from algebraic geometry (see Appendix, Proposition A.1.1) according to which in an algebraic variety defined over a separably closed field k^s the points defined over k^s form a Zariski dense subset; indeed, such a subset is infinite if the variety has positive dimension.

2.2 Splitting Fields 35 Corollary 2.2.6 A finite dimensionalk-algebraAis a central simple algebra if and only if there exist an integer n >0 and a finite Galois field extension K|k so that A⊗^kK is isomorphic to the matrix ring Mn(K).

Proof: This follows from Theorem 2.2.1, Proposition 2.2.5 and the well-known fact from Galois theory according to which every finite separable field extension embeds into a finite Galois extension.

Remarks 2.2.7

1. It is important to bear in mind that if A is a central simple k-algebra of degree n which does not split over k but splits over a finite Galois extensionK|kwith groupG, then the isomorphismA⊗kK ∼=Mn(K) is notG-equivariant if we equipM_n(K) with the usual action ofGcoming from its action on K. Indeed, were it so, we would get an isomorphism A∼=Mn(k) by taking G-invariants.

2. In traditional accounts, Proposition 2.2.5 is proven by showing that the separable splitting field can actually be chosen among the field extensions ofkthat arek-subalgebras ofA. We shall prove this stronger fact later in Proposition 4.5.4. However, it is not always possible to realize a Galois splitting field in such a way, as shown by a famous counterexample by Amitsur (see Amitsur [2] or Pierce [1]; see also Brussel [1] for counterexamples over Q(t) and Q((t))). Central simple algebras containing a Galois splitting field are called crossed products in the literature.

We finally discuss a method for finding Galois splitting fields among k-subalgebras of A. The basic idea is contained in the following splitting cri-terion, inspired by the theory of maximal tori in reductive groups.

Proposition 2.2.8 A central simple algebra A of degree n over a field k is split if and only if it contains a k-subalgebra isomorphic to the direct product kⁿ=k× · · · ×k.

For the proof we need a well-known property of matrix algebras.

Lemma 2.2.9 The k-subalgebras in Mn(k) that are isomorphic to kⁿ are conjugate to the subalgebra of diagonal matrices.

Proof: Giving a k-subalgebra isomorphic to kⁿ is equivalent to specifying n elements e₁, . . . e_n that form a system of orthogonal idempotents, i.e. sat-isfy e²_i = 1 for all i and eiej = 0 for i 6= j. Identifying Mn(k) with the endomorphism algebra of ann-dimensional k-vector spaceV, we may regard the e_i as projections to 1-dimensional subspaces V_i in a direct product de-composition V =V1 ⊕ · · · ⊕Vn of V. Choosing a vector space isomorphism V ∼= kⁿ sending Vi to the i-th component of kⁿ = k⊕ · · · ⊕k gives rise to the required conjugation.

Proof of Proposition 2.2.8: Of courseMn(k) contains subalgebras isomor-phic tokⁿ, whence the necessity of the condition. Conversely, assume given a k-algebra embedding i : kⁿ → A, and let e1, ..., en be the images of the standard basis elements ofkⁿ. By Rieffel’s Lemma it will be enough to show thatAe1 is a simple leftA-module and that the natural mapk →EndA(Ae1) is an isomorphism. Ifk is algebraically closed, thenA ∼=M_n(k) by Corollary 2.1.7. Lemma 2.2.9 then enables us to assume thatiis the standard diagonal embedding kⁿ ⊂ Mn(k), for which the claim is straightforward. If k is not algebraically closed, we pass to an algebraic closure and deduce the result by dimension reasons.

Corollary 2.2.10 Let A be a central simplek-algebra containing a commu-tativek-subalgebraK which is a Galois field extension ofk of degreen. Then K is a splitting field for A.

Proof: By Galois theory, theK-algebra K⊗^kK is isomorphic to Kⁿ (see the discussion after the statement of Lemma 2.3.8 below), and thus is a K-subalgebra of A⊗kK to which the proposition applies.