Primitive elements of abelian extensions and fields generated by polygon diagonals

Abstract

In this thesis we are interested in primitive elements of abelian extensions of Q. By the Kronecker-Weber Theorem we know that abelian extensions are subfields of cyclotomic extensions Q(ζ_n). For the case where n = p^a with p an odd prime, we will show that the primitive elements of these field extensions are traces of powers of ζ_n over a subgroup H of the Galois group. In the last part of this thesis, we will discuss field extensions that are generated by the ratio of the lengths of two diagonals of a regular polygon. More specifically, we will discuss an article concerning this subject and show that there are some false claims in the article.