Modules with Integral Discriminant Matrix

Author: Maurer, Donald Eugene

Year: 1969

Degree: Dissertation (Ph.D.)

Advisor: Taussky-Todd, Olga

Committee Member: Unknown, Unknown

Option: Mathematics

DOI: 10.7907/BQG6-4P65

Abstract

Let F be a field which admits a Dedekind set of spots (see O'Meara, Introduction to Quadratic Forms) and such that the integers ZF of F form a principal ideal domain. Let K|F be a separable algebraic extension of F of degree n. If M is a ZF-module contained in 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, with larger determinant, by an integral transformation.

These two main results are then applied to characterize normal algebraic extensions K of the rationals in which ZK is maximal with respect to having an integral discriminant matrix.

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