Publication Date


Document Type


First Advisor

Geline, Michael

Degree Name

Ph.D. (Doctor of Philosophy)

Legacy Department

Department of Mathematical Sciences


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.


125 pages




Northern Illinois University

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