#### Thesis Title

A Constructive Approach to the Inverse Galois Problem Over the Rationals

#### Date of Graduation

Summer 2005

#### Degree

Master of Science in Mathematics

#### Department

Mathematics

#### Committee Chair

Cameron Wickham

#### Abstract

The inverse Galois problem of Galois Theory asks: Given a group G, is there a field extension of a field k, such that the extension’s Galois group is G? In this thesis, we are concerned with a constructive method of finding a Galois extension over the field Q of rational numbers, given in advanced a finite group. In the first two parts we expand on methods of realizing Galois extensions found in Dummit and Foote’s Abstract Algebra. First we show that cyclic groups can be realized as Galois groups by considering the cyclotomic extensions of Q. We then examine the more general case of abelian groups. Next, we explain how a finite symmetric group S can be realized as a Galois group by constructing an irreducible polynomial with rational coefficients whose Galois group is S. The last part implements a method found in an article by Richard A. Dean published in The American Mathematical Monthly. In the article Dean finds the Galois extension of the quaternion group of order 8. We then try to extend the method to the quaternion group of order 16.

#### Keywords

inverse Galois Theory, Galois Theory, group theory, field extension, ring theory

#### Subject Categories

Mathematics

#### Copyright

© Branden R. Stone

#### Recommended Citation

Stone, Branden R., "A Constructive Approach to the Inverse Galois Problem Over the Rationals" (2005). *MSU Graduate Theses*. 1619.

https://bearworks.missouristate.edu/theses/1619

**Dissertation/Thesis**