10 1. A DIVISIO N ALGEBR A I S A CENTRA L SIMPL E ALGEBR A

Of course , divisio n algebra s aris e naturall y becaus e of Wedderburn's Theore m

about centra l simpl e algebras . W e recall a version of this theore m below .

THEOREM

1.3. (e.g., [35] p. 200-205) Any central simple algebra A/F has

the form M r(D) where D/F is a division algebra and Mr(D) is the algebra ofrxr

matrices over D. Conversely, any algebra of the form M r(D) for a division algebra

D/F is a central simple algebra over (i.e., with center) F. Furthermore, if A is

central simple then any A module is the direct sum of copies of a unique (up to

isomorphism) irreducible module.

Of course if F is algebraically closed then Lemma 1.2 and Theorem 1.3 combine

to give :

COROLLARY 1.4. If A/F is a central simple algebra, and F is algebraically

closed then A = M n(F).

Let u s quote som e important fact s abou t centra l simpl e algebras , with a refer -

ence. W e omit an y proofs o r explanation becaus e w e will revisi t thes e result s i n a

more general for m in section two. Note , however, tha t th e proofs ther e will assum e

1.5 below. Wha t i s important abou t 1.5 is that th e dass o f central simpl e algebra s

are close d unde r bas e extension and tensor product . Thi s is quite false for division

algebras an d a primary reaso n divisio n algebra s ar e always studie d i n the context

of centra l simpl e algebras .

PROPOSITION 1.5. (e.g., [35] p. 219)

a) Let A/F, B/F be central simple algebras and K D F a field extension. Then

A g)F B is a central simple algebra with center F and A ®F L is a central

simple algebra with center L.

b) Conversely, if A is an F algebra and A ®F L is central simple over L then

A/F is central simple.

In particular, i f A/F i s central simple , and F D F is an algebraic closure, the n

by a ) above, 1.2 and 1.3, A ®p F = M n(F) fo r some n. W e say that ever y centra l

simple algebr a i s a form o f matrices meanin g tha t i t becomes matrice s afte r bas e

extension to an algebraically close d field. Thi s has a large number of consequences.

The first, an d easiest, i s that th e dimension o f a central simpl e algebr a A/F (ove r

F) i s always o f the form n

2.

W e call n th e degree o f A an d write i t a s deg(A).

Furt her consequence s ar e explicitly give n i n sectio n four , bu t in fac t th e whol e

subject ca n be recharacterized a s describing al l forms o f matrices.