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
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