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

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.
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