#### Date of Award

2017

#### Degree Type

Thesis

#### Degree Name

Honors Thesis

#### Department

Mathematics

#### First Advisor

Chris Wilson

#### Second Advisor

William Johnston

#### Abstract

Let F be the field of fractions of R, a ring of power series with coefficients in some field. Let K/F be a finite Galois extension, and assume the integral closure S of R in K is also a power series ring.

In this paper, we consider a construction from ring theory called a crossed product algebra and an associated function f: G × G →K called a cocycle that is used to define a multiplication operation for the crossed product algebra. Consider a crossed product algebra whose cocycle f takes values in S. We give a proof concerning which of f’s values must be invertible in S when G is dihedral. We apply the results of this proof to create an algorithm, which computes the values in the cocycle table. We consider the question of which values the cocycle f can take and thus learn about the possible multiplicative structures for crossed product algebras arising from dihedral field extensions.

#### Recommended Citation

Lee, Kaitlyn, "Crossed Product Algebras over Dihedral Field Extensions" (2017). *Undergraduate Honors Thesis Collection*. 393.

https://digitalcommons.butler.edu/ugtheses/393