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 : 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
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