Modules with Integral Discriminant Matrix
<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<sub>F</sub> 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<sub>F</sub&...
| Summary: | <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<sub>F</sub> 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<sub>F</sub>-module contained
in K, and σ<sub>1</sub>, σ<sub>2</sub>, ..., σ<sub>n</sub> is a Z<sub>F</sub>-basis for M, the matrix D(σ) = (trace<sub>K|F</sub>(σ<sub>i</sub>σ<sub>j</sub>)) 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<sub>ij</sub> a<sub>ij</sub>x<sub>i</sub>x<sub>j</sub> (a<sub>ij</sub> = aa<sub>ji</sub>),
<p>with integral matrix (a<sub>ij</sub>) 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<sub>K</sub> is maximal with
respect to having an integral discriminant matrix.</p> |
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