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

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.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2014

Author dc:creator

• Carter, James Edgar

Contributors dc:contributor

• McCulloh, L.,

• Mathematics

Identifier

(UMI)AAI9305482

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/72532