Date

2017

Document Type

Dissertation

Degree

Doctor of Philosophy

Department

Mathematics

First Adviser

Dodson, Bruce

Other advisers/committee members

Davis, Donald; Weintraub, Steven; Dougherty, Steven

Abstract

Given a CM field $K$ of degree $2n$, there is a triple $(G,H,\rho)$ called an imprimitivity structure in which $G$ is the Galois group of the Galois closure of $K$, $H$ is the subgroup of $G$ that fixes $K$, and $\rho \in G$ is a distinguished central order $2$ element that is induced by complex conjugation. Dodson showed in \cite{Dodson} that imprimitivity structures may be identified under the action of $G$ into equivalence classes called $\rho$-structures. He determined all possible $\rho$-structures for $n = 3, 4, 5$ and $7$ and a partial list for $n = 6$. In \cite{Zoller}, Zoller completed the list of $\rho$-structures for $n = 6$, produced complete lists for $n = 8, 9,$ and $10$, and laid the foundation for the completion of $n = 12$. In the present investigation we continue the study for $n=12$ as well as begin the study for $n=14$ and $15$. Nondegenerate CM types have rank $n + 1$ and make up the majority of CM types for a given CM field. In contrast, degenerate CM types have rank less than $n + 1$ and occur less frequently. Dodson was concernedwith the identification of degenerate CM types and their relationship with CM fields that contain an imaginary quadratic subfield. Zoller extended this investigation from $n = 6$ to $n = 8, 9, 10,$ and $12$. He also found and characterized degenerate CM types arising from a CM field that do not contain an imaginary quadratic subfield for $n=8,9,$ and $10$. In the case $n=12$ he determined the types of CM fields that contain an imaginary quadratic subfield. We continued the study in degree $2n=24$ by looking at $\rho$-minimal groups and using a subgroup analysis. The difficulty in getting a complete picture is that there are over $19,\! 000$ Galois groups of CM fields in degree $24$. So we use these special methods along with Zoller's study of CM fields containing an imaginary quadratic subfield to find bounds on the number of degenerate types. For $n=14$, we find and characterize degenerate CM types arising from a CM field that may or may not contain an imaginary quadratic subfield. For the larger order Galois groups in degree $2n=28$, we adapt the subgroup analysis from degree $24$ to lessen computation time. For $n=15$, we begin to find and characterize degenerate CM types arising from a CM field that may or may not contain an imaginary quadratic subfield. We find similar results to the $n=9$ case.

Included in

Mathematics Commons

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