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Modules with Integral Discriminant Matrix


Maurer, Donald Eugene (1969) Modules with Integral Discriminant Matrix. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BQG6-4P65.


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|Fiσj)) 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.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:(Mathematics)
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Taussky-Todd, Olga
Thesis Committee:
  • Unknown, Unknown
Defense Date:7 April 1969
Funding AgencyGrant Number
Record Number:CaltechTHESIS:03282017-155524180
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:10114
Deposited By: Benjamin Perez
Deposited On:29 Mar 2017 14:32
Last Modified:03 May 2024 20:40

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