University of Illinois Urbana-Champaign

An Equal-Distribution Result for Galois Module Structure

Foster, Kurt Christopher

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Description

Title
An Equal-Distribution Result for Galois Module Structure
Author(s)
Foster, Kurt Christopher
Issue Date
1987
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 K be a number of field with ring of integers o, and let G be a fixed finite group. If K$\sb\pi$ is a tame Galois G-extension, the integral closure ${\cal O}\sb\pi$ of o in K$\sb\pi$ is a locally free rank one oG-module, so realizes a class cl(${\cal O}\sb\pi$) in the locally free class group Cl(oG). We let R(oG) denote the set of classes so realized. In the case where G is an elementary abelian group, we obtain the following result.
  • Theorem. Let K be a number field with ring of integers o, and G an elementary abelian group. Let c $\in$ R(oG), and denote by N(c,X) the number of tame Galois G-extensions K$\sb\pi$ for which cl(${\cal O}\sb\pi$) = c and having absolute discriminant ${\rm d}({\cal O}\sb\pi/\doubz) \le {\rm X}.$ Then N(c,X) $\sim\beta$ $\cdot$ Y(log Y)$\sp{\rm r-1}$ where Y$\sp{\varphi(\vert{\rm G}\vert)}$ $\cdot$ d(o/$\doubz$)$\sp{\vert{\rm G}\vert}$ = X. Here, $\beta$ is a positive constant depending on K and G, but not on the class c $\in$ R(oG), and r is a positive integer which depends only on K and G.
  • This is proved in Theorem (4.1). It tells us that the number of tame Galois G-extensions K$\sb\pi$ for which d(${\cal O}\sb\pi/\doubz) \le$ X and cl(${\cal O}\sb\pi$) = c $\in$ R(oG) is asymptotically the same for each c $\in$ R(oG). This is the equal distribution result of the title.
  • A result for fields is retrieved by noting that, for a tame Galois field extension L/K with Gal(L/K) = $\Gamma$ isomorphic to G, a choice of isomorphism $\Gamma \cong {\rm G}$ makes L into a G-extension. This is described in Chapter 4, and the result given by Theorem (4.2).
Graduation Semester
1987
Type of Resource
text
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http://hdl.handle.net/2142/71253

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