32 M. L. RACINE
(i) h(ax, y) = ah(x, y)
(ii) h(y, x) = -h(x, y).
A matrix A e $ is skew-hermitian if A = -A.
n
In this section we restrict ourselves to K a complete discrete v a l u a -
tion field. Let ($, -) be a finite dimensional central associativ e division
algebra with involution over K. The center of $, C($) is either K or a
separable quadratic field extension of K ([1], p. 153). Let N be the
reduced norm of £/C(JG) and N(a) = N
n W i r
(N (a)), a

fl; N(ora) = » m N(a)
u(^)/K r
i /
a
e
K, a e #, m the degree of $/K. Consider the map a - | N(a)| from
& to the non-negative reals . Since | N(a)| = \ a | = | or | , or

K, we
i /
denote this map | | . Therefore | a | = | N(a)| extends the valuation | |
on K. Arguing a s in the commutative cas e it is eas y to show that | | is a
valuation on & ([37] , 14. 1) and that $ is complete under this valuation
([37], 11.7) . The algebra & ha s a unique maximal order C = {a &| | a | 1}
= {a &| N(a) o} ([6], p. 100). Since | | is discrete on K, | | is d i s -
crete on & and f = {a f !0| | a | 1} the prime ideal of © is principal,
¥ = pO. Let v(a) be the exponential valuation of a i . e . v(0) = «, v(a) = i
if | a | = | p X | , i

Z . . Denote K - {0} by K, ^ - {0} by i . Let
u. (A) e K[?\] be the minimum polynomial of a over K. Then u. (a) = u. (a) = 0
' a ^ a a
and since ±N(a) is a power of the constant term of | ± (7\), N(a) = N(a).
a
Therefore | a| = | a | . Consider V(M(J6, -)) . Since aa
e
M(&, -) and
v(pLpL) = 21, v(M(fl,-)) D 2Z. If d

H ( f l , - ) with v(d) = 2r + 1 then
p " r d p " r

W(fl, - ) . So v(p~rdp " r ) = 1 and v(H(i, -)) = Z . Therefore
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