Steinitz Classes of Tamely Ramified Nonabelian Extensions of Algebraic Number Fields of Degree P(3)

Carter, James Edgar

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Description

Title

Steinitz Classes of Tamely Ramified Nonabelian Extensions of Algebraic Number Fields of Degree P(3)

Author(s)

Carter, James Edgar

Issue Date

1992

Doctoral Committee Chair(s)

McCulloh, L.,

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

Mathematics

Abstract

Let L/k be a finite extension of algebraic number fields of degree n with rings of integers ${\cal D}\sb{L}$ and ${\cal D}\sb{k}$. As an ${\cal D}\sb{k}$-module ${\cal D}\sb{L}$ is finitely generated and torsion free and thus can be written as a direct sum of $n - 1$ copies of ${\cal D}\sb{k}$ and a fractional ideal ${\cal J}$ of k. The class $c\ell({\cal J})$ of ${\cal J}$ in the ideal class group C(k) of k is the Steinitz class C(L,k) of ${\cal D}\sb{L}$. Now let G be a finite group of order n. As L varies over all normal extensions of k such that Gal(L/k) $\simeq$ G, C(L,k) varies over a subset of C(k). These are the realizable classes of k with respect to G which we denote by R(k,G). If we consider only tamely ramified extensions then we denote this set by $R\sb{t}(k,G)$. If G is an abelian group with exponent b, and k contains the multiplicative group of b-th roots of unity, then it is known that $R\sb{t}(k,G)$ = $C(k)\sp{m}$ (the subgroup of $C(k)$ consisting of m-th powers of elements of $C(k))$ where m is a positive rational integer which depends on the structure of G. We obtain a partial generalization of this result in the sense that if n = $p\sp3$ where p is an odd prime then we can remove the restriction that G be an abelian group.

Graduation Semester

1992

Type of Resource

text

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