1.2. SIMPLICITY, ISOTYPICITY, AND ENDOMORPHISM ALGEBRAS 17

1.2.2.1. Definition. A central simple algebra over a field K is a non-zero asso-

ciative K-algebra of finite dimension such that K is the center and the underlying

ring is simple (i.e., has no non-trivial two-sided ideals).

A central division algebra over K is a central simple algebra over K whose

underlying ring is a division algebra.

Among the most basic examples of central simple algebras over a field K are

the matrix algebras Matn(K) for n 1. The most general case is given by:

1.2.2.2. Proposition (Wedderburn’s Theorem). Every central simple algebra D

over a field K is isomorphic to Matn(Δ) = EndΔ(Δ⊕n) for some n 1 and some

central division algebra Δ over K (where Δ⊕n is a left Δ-module). Moreover, n is

uniquely determined by D, and Δ is uniquely determined up to K-isomorphism.

Proof. This is a special case of a general structure theorem for simple rings; see

[53, Thm. 4.2] and [53, §4.4, Lemma 2].

In addition to matrix algebras, another way to make new central simple algebras

from old ones is to use tensor products:

1.2.2.3. Lemma. If D and D are central simple algebras over a field K, then so

is D ⊗K D . For any extension field K /K, DK := K ⊗K D is a central simple

K -algebra.

Proof. The first part is [53, §4.6, Cor. 3]; the second is [53, §4.6, Cor. 1, 2].

1.2.3. Splitting fields. It is a general fact that for any central division algebra

Δ over a field K, ΔKs is a matrix algebra over Ks (so [Δ : K] is a square). In

other words, Δ is split by a finite separable extension of K. There is a refined

structure theory concerning splitting fields and maximal commutative subfields of

central simple algebras over fields; [53, §4.1–4.6] gives a self-contained development

of this material. An important result in this direction is:

1.2.3.1. Proposition. Let D be a central simple algebra over a field F , with

[D : F ] =

n2.

An extension field F /F with degree n embeds as an F -subalgebra of

D if and only if F splits D (i.e., DF Matn(F )). Moreover, if D is a division

algebra then every maximal commutative subfield of D has degree n over F .

Proof. The first assertion is a special case of [53, Thm. 4.12]. Now assume that

D is a division algebra and consider a maximal commutative subfield F . In such

cases F splits D (by [53, §4.6, Cor. to Thm. 4.8]), so n|[F : F ] by [53, Thm. 4.12].

To establish the reverse divisibility it suﬃces to show that for any central simple

algebra D of dimension

n2

over F , every commutative subfield of D has F -degree

at most n. If A is any simple F -subalgebra of D and its centralizer in D is denoted

ZD(A) then

n2

= [A : F ][ZD(A) : F ] by [53, §4.6, Thm. 4.11]. Thus, if A is also

commutative (so A is contained in ZD(A)) then [A : F ] n.

The second assertion in Proposition 1.2.3.1 does not generalize to central simple

algebras; e.g., perhaps D = Matn(F ) with F having no degree-n extension fields.

In general, for a splitting field F /F of a central simple F -algebra D, the

choice of isomorphism DF Matn(F ) is ambiguous up to composition with the