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