Citation
Maurer, Donald Eugene (1969) Modules with Integral Discriminant Matrix. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BQG64P65. https://resolver.caltech.edu/CaltechTHESIS:03282017155524180
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 Z_{F} of F form a principal ideal domain. Let KF be a separable algebraic extension of F of degree n. If M is a Z_{F}module contained in K, and σ_{1}, σ_{2}, ..., σ_{n} is a Z_{F}basis for M, the matrix D(σ) = (trace_{KF}(σ_{i}σ_{j})) 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 = Σ_{ij} a_{ij}x_{i}x_{j} (a_{ij} = aa_{ji}),with integral matrix (a_{ij}) 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 Z_{K} 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): 
 
Thesis Committee: 
 
Defense Date:  7 April 1969  
Funders: 
 
Record Number:  CaltechTHESIS:03282017155524180  
Persistent URL:  https://resolver.caltech.edu/CaltechTHESIS:03282017155524180  
DOI:  10.7907/BQG64P65  
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  10114  
Collection:  CaltechTHESIS  
Deposited By:  Benjamin Perez  
Deposited On:  29 Mar 2017 14:32  
Last Modified:  03 May 2024 20:40 
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