One final feature of this algebraic computation is the explicit
nature of the occurrence of an element of order four in C (Z) in the
symmetric (e = +1) case. Since the groups C^(Z) have no four-
torsion, any element of order four must have a non-trivial coupling
invariant. We give an explicit algebraic example of this phenomena in
Section Four is the algebraic core of this paper. In this section
we relate the domain S which is an order in its quotient field
E to the maximal order D , the Dedekind domain of all algebraic
integers in E . We first introduce the ring C (Z) , which is the
group of equivalence classes of isometric structure whose underlying
module structure can be lifted to D in a manner so that the endomor-
phism t corresponds to multiplication by an element x in D which
is primitive in the field extension E/Q . We can make a similar defini-
tion for any order S in E and obtain Cq (2Z) . Note that C (Z) is
identical with C (Z) for the order S = Z [X]/(cp(X)) . The isometric
structure condition tx,y = x,(1 - t)y implies that the minimal
polynomial c p (X) of t must satisfy c p (x) = _+cp(l-x) and therefore the
requirement X = 1 - X induces an involution of the order S and hence
E . The main algebraic idea is the discovery of a method to relate non-
singular e-symmetric isometric structures over the integers to certain
non-singular e-Hermitian forms over S . The first component of this
relationship was highlighted by Milnor in his germinal paper on n0n
isometries of inner product spaces", [Ml]. Here he shows how to use
the natural bilinear form over a field, the trace form, to relate
isometries of inner produce spaces over a field to a Hermitian form over
a finite extension field to which the F[X]-module structure determined
by t can be reduced. This popular lemma is given in Lemma 2.6. The
second technical feature of our computation is the natural occurrence of
the inverse different of an order S , A (S/Z) = {x in E | Trace ,
(sx) is an integer for all s in s} . There is a natural non-singular
pairing with S given by the trace form on E . The trace lemma then
allows a natural identification of non-singular e-Hermitian forms on a
torsion-free S-module with values in A (S./Z) with integral non-singular
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