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.