ARITHMETICS OF JORDAN ALGEBRAS 15
2
Since x = 1U , (iii) = (iv). Finally to show that (iv) = (i) we may assum e
2
that K is a complete discrete valuation field. By (7), N(x ) = N(1U ) =
2 2
N(l)N(x) = N(x) . Consider the map x - | N(x)| , where | | is the valua -
tion on K. Now M is a finitely generated o-module. Let {x.} , 1 i r,
be a set of generators of M,JJL= max { | N(x. )| , | N(x., x,)| }. Let or, p

o .
l i , j r l l J
Since | o | 1 , |N(*x + Py)| max{ \a\2 |N(x)| , \a\ |P||N(x, y)| , |P| 2 |N(y)| }
max{ |N(x)| , |N(x,'y)|, | N ( y ) | } . Therefore | N ( M ) | , i . Either |j. = |N(x.)| for
some i or \i - |N(x., y.) | |N(x.) |, |N(x.) | for some i, j and |N(x. +x.) | =
|N(x.,x.)| . In both c a s e s x -* |N(x)| attains its maximum on M. Assume
(iv); sinc e | N ( x 2 ) | = | N ( x ) 2 | = | N ( x ) | 2 , |N(x)| must be 1 for x
6
M .
Therefore N(M) c o.
q. e. d.
COROLLARY 1. The maximal orders of # = P(N, 1) are the maximal
lattices containing 1 on which N is integral.
COROLLARY 2. Any order of # = #(N, 1) is contained in a maximal
order.
We wish to consider the isomorphisms of maximal orders when K is
a complete discrete valuation field. In that cas e any two lattice s of ^ on
which N is integral and which are maximal with respect to that property are
isometric ([3 7], 91. 2 p. 240). Let L be a fixed such lattice , 0 = 0 ( & N )
the orthogonal group of the quadratic spac e (p, N), O(L) = {creOlLc = L} .
Let C - {x * p\ N(x) = 1}; C D L ^ 0 sinc e L is isometric to a lattic e con -
taining 1. Let [x] denote the orbit under O(L) of the element x C n L.
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