ARITHMETICS OF JORDAN ALGEBRAS

33

v(M(fl,-)) = 2Z or Z . If v(M(A,-)) = Z we may assum e p

€

H(fl,-) and we

will sa y that & ha s a symmetric prime. Otherwise v(ii($, -)) = 222 and we

sa y that & ha s no symmetric prime.

We start with a few well-known lemmas and then give some result s of

Hijikata [10] (see als o [9]) on orders in associativ e algebras with involution

over a complete discrete valuation field.

LEMMA 7. An arbitrary invertible A

€

& may be written UB, U a

n

unit of D ,

n

12

r. € Z , b. . € & and v(b

i t

) r, or b, , = 0.

i ij u J u

PROOF. ([6], Chapter VI, §11)

LEMMA 9. Let A be an invertible element of & then there exist s

n

U . , U

0

, units of £ , such that

i L n

U 1 A U 2 =

S. 6 Z , S S . . . S