ARITHMETICS OF JORDAN ALGEBRAS

COROLLARY. If ?/$ is a finite dimensional central Jordan division

p

algebra then ^ is a form of an algebra of type (2), (3) or (4) ( i . e . 9 is of

type (2), (3) or (4) for some field extension P of $).

PROOF. As noted above we may assum e that char. $ = 2, in which

cas e we give the following proof due to McCrimmon. Let P be a Galois

P

extension of $» By Lemma 2 of [3 0] 9 {S semisimple and P is the centroid

P P

of 9 . Since 9 is finite dimensional, so is 9 which therefore satisfies

P

DCC on principal inner ideals . Hence by Proposition 4 of [3 0] 9 is simple.

If m (7\) is not purely inseparable for some x e ^ then P is chosen to con -

p

tain a root of m ("A). Hence 9 is no longer a division algebra and so must

2 n

be of type (2), (3) or (4). If all x are purely inseparable, m (A) = A - x(x)

and $ is infinite. In particular, if Q, is the algebraic closure of $,

?

n

o o

x € Ql for all x e 9 . Thus all elements of 9 are invertible or

nilpotent. So 9 /®($ ) is a division algebra (which must therefore be ftl,

[17] p . 3.61) or a traceles s 9(Q,1) (with x

e

fil)[3 5]. But the only

non degenerate traceles s 2(Q, 1) over ft is ftl. The canonical map

9~* 9 -+ 2 IQ(9 ) maps 1 into 1, so by the simplicity 2 it is infective.

Therefore 9 is isomorphic to a $-subalgebr a of fil. Let x, y

e

9»

2

yU = x y. It is eas y to se e that U belongs to the centroid $ of 9, so

x x

2 2

x

6

$. Therefore ^ = $1 + $x + . . . + $x , where x. = ^. € $ are linearly

2

independent over $ (J is a division algebra) and yU = Q(x)y, where if

0

n

0 2 V " 2

x = *01 + «

1

x

1

+ . . . + *

n

x

n

Q( x ) = *

0

+ I a

i

^{- Hence 9 is of type (2)

i=l

and this completes the proof.