ARITHMETICS OF JORDAN ALGEBRAS
17
x
e
C. But C n L = U xO(L). Since C n L is compact
xcCnL
r
c n L = U X.CXD, x.

c n L.
i=l
l l
q. e. d.
Knebusch [24] has studied the orbits [x], x
e
C n L when the
characteristic of K is not 2.
§4. Maximal orders of Jordan division algebras over a complete discrete
valuation field.
Let #/K be a central Jordan division algebra, K a complete discret e
valuation field.
LEMMA 2. An element x

? is integral if and only if N(x)
e
o.
PROOF. Consider K(x), x ^ 0; this is a field. Since x is invertible
N(x) £ 0. Lemma 2 is then only a restatement of the same well known result
for fields (e.g . [6], p. 99; [40], p. 220).
q. e. d.
PROPOSITION 8. Let ^ be a Jordan division algebra over K a com -
plete discrete valuation field. The set of elements with integral norm is the
unique maximal order of ^.
PROOF. Let M = {x
6
p\ N(x) = o}. Since N(l) = 1, 1 e M; als o
2
N(x), N(y) o imply N(xU ) = N(x)N(y) o so xU eM. We wish to show
next that N(x), N(y) e o imply N(x + y)

o. Without los s of generality we
(x"
1
)
may assume | N(y)| | N(x)| where | | is the valuation on K. Let f '
be the x " isotope of p. Since N
( x
^(z) = N(x)
- 1
N(z), N
( x
\z) fi 0 for
Previous Page Next Page