Modules with Integral Discriminant Matrix

• Main Author: Maurer, Donald Eugene
• Format: Others
• Published: 1969
• Online Access: https://thesis.library.caltech.edu/10114/1/Maurer_DE_1969.pdf; Maurer, Donald Eugene (1969) Modules with Integral Discriminant Matrix. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/BQG6-4P65. https://resolver.caltech.edu/CaltechTHESIS:03282017-155524180 <https://resolver.caltech.edu/CaltechTHESIS:03282017-155524180>

<p>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 K|F be a separable algebraic extension of F of degree n. If M is a Z_F-module contained in K, and σ₁, σ₂, ..., σ_n is a Z_F-basis for M, the matrix D(σ) = (trace_K|F(σ_iσ_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</p> f = Σ_ij a_ijx_ix_j (a_ij = aa_ji), <p>with integral matrix (a_ij) can be obtained from another such form, with larger determinant, by an integral transformation.</p> <p>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.</p>