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 ,
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