Let F be a field which admits a Dedekind set of spots (seeO'Meara, Introduction to Quadratic Forms) and such that the integersZF of F form a principal ideal domain. Let K|F be a separablealgebraic extension of F of degree n. If M is a ZF-module containedin K, and σ1, σ2, ..., σn is a ZF-basis for M, the matrix D(σ) = (traceK|F(σiσj)) is called a discriminant matrix. We study modules which have an integral discriminant matrix. When F is the rational field, we are able to obtain necessary and sufficient conditions on det D(σ) in order that M be properly contained in a larger module having an integral discriminant matrix. This is equivalent to determining when the corresponding quadratic form
f = Σij aijxixj(aij = aaji),with integral matrix (aij) can be obtained from another such form, withlarger determinant, by an integral transformation.
These two main results are then applied to characterize normalalgebraic extensions K of the rationals in which ZK is maximal withrespect to having an integral discriminant matrix.
