16 M . L. RACINE

PROPOSITION 6. Let & K, L be as above. Then the isomorphism

c l a s s e s of maximal orders of P are in one-to-on e correspondence with the

orbits [x], x c C n L.

PROOF. Let M. i = 1,2 be maximal orders of $9 cp. isometries of

M, onto L. To M. associat e the orbit \1P.]. If \b. is another isometry

I I

L iJ

l

of M, onto L then since 1 + . = 1P.P. +. and w, +. eO(L) we have [1 cp.] -

I I

1 1 l 1 1

iJ

[1 \\i.]. If [1 p ] = [1 cp ] then let a e O(L) be such that 1 p a = 1 cp .

Hence p a cp is an isometry of M onto M and 1 cp a cp = 1. So

by Proposition 4 M and M are isomorphic (as Jordan algebras over o).

Finally given [x], there exists a ^ O with xcp = 1 (by Witt's Theorem).

Then M = Lcp is a maximal order of # associate d to [x],

q. e. d.

A complete discrete valuation field K is local if its residue field

o/p is finite. The quotient field of a Dedekind ring is global if every

completed localization is local.

PROPOSITION 7. If K is local the number of orbits [x], x

€

C n L is

finite.

PROOF. Since C = N~ (1), it is closed. But L is compact. There-

fore C nL is compact. Let x be an arbitrary fixed element of C, 0(9 x ) =

{a € 0 | xa - x } . Since O acts transitively on C, C is homeomorphic to

O/Qk$9 x) and the map p -* xp from O to C is open ([38], p. 148). Since

0(L) is open and x was arbitrarily chosen, xO(L) is open in C for any ,