#### Publication Date

2022

#### Document Type

Dissertation/Thesis

#### First Advisor

Geline, Michael

#### Degree Name

Ph.D. (Doctor of Philosophy)

#### Legacy Department

Department of Mathematical Sciences

#### Abstract

This dissertation examines conductors for maximal orders of group rings, $FD$, contained in integral group rings, $\mathcal{O}_F D$, where $D$ is a finite group, $F$ a local field, and $\mathcal{O}_F$ its discrete valuation ring. (Given two rings $R \subseteq S$, the conductor of $S$ in $R$, if it exists, is the largest ideal of $S$ contained in $R$.) First, we use a theorem of Jacobinski to make this conductor explicit where $F$ is a particular extension of $\mathbb{Q}_p$ of varying ramification index and $D$ is any elementary abelian $p$-group. Next, we examine this conductor problem for this same $F$ within two components of $FD$, where $D$ is an elementary abelian group of order $p^2$. In particular, we aim to find the conductor for the maximal order of $FDe_1 \oplus FDe_2$ within $\mathcal{O}_F D(e_1 + e_2)$, where $e_1$ and $e_2$ are idempotents of $FD$ corresponding to nontrivial characters of $D$. The solution to this problem is determined for $p \in \{2, 3, 5\}$ and a conjecture for all $p$ is presented. Applications to the classification of $\mathcal{O}_F D$-lattices are discussed.

#### Recommended Citation

Randazzo, Brooke, "Conductors for Maximal Orders of Group Rings over Local Fields" (2022). *Graduate Research Theses & Dissertations*. 7580.

https://huskiecommons.lib.niu.edu/allgraduate-thesesdissertations/7580

#### Extent

125 pages

#### Language

eng

#### Publisher

Northern Illinois University

#### Rights Statement

In Copyright

#### Rights Statement 2

NIU theses are protected by copyright. They may be viewed from Huskie Commons for any purpose, but reproduction or distribution in any format is prohibited without the written permission of the authors.

#### Media Type

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