Galois theory and number theory

The focus of this thesis is to use Galois Theory to prove results in Number Theory. As a result, we begin by familiarizing ourselves with some significant results in Galois Theory. We then proceed to discuss other pertinent results in Algebraic Number Theory. This discussion provides the tools necessary to prove key theorems later in the text. The first five chapters, along with Chapters 8 and 10, are included to provide the reader with background information. As a result, the reader will find that not every result in these chapters is followed by a proof. The first main result we discuss in full is Kummer's Theorem. This theorem connects factoring a polynomial mod p with factoring p once it has been lifted to the ring of integers. A proof of the well known Quadratic Reciprocity Law follows which incorporates Kummer's Theorem and results in Galois Theory. The remaining text is devoted to the Kronecker-Weber Theorem which states that every Abelian extension of Q is contained in a cyclotomic extension. We conclude the discussion of the Kronecker-Weber Theorem with a few concrete examples.