Radical extensions and Galois groups

Abstract

The use of symbolic computing is one of the characteristics of a computer algebra package. For example, the number square root of 2 is represented as a symbol with the property that its square is 2. This enables us to do exact calculations. However, the computer algebra package often give complicated expressions although there is a simpler expression for the number that we are interested in. The simplification of nested radicals was an important motivation for the research in this thesis. Given a nested radical alpha we can build a chain of field extensions, such that the largest field in this chain is a Galois extension containing alpha and such that all subextensions in the chain are generated by a radical. Galois theory gives a powerfull tool to study field extensions. In the article `Simplifications of nested radicals' Susan Landau uses Galois theory to provide an algorithm that calculates a simple expression for alpha. In all publications about nested radicals roots of unity play an important role. In Landau's algorithm roots of unity cause problems in proving that the algorithm gives the simplest possible representation as a nested radical. In chapter 1 of this thesis we study Galois extensions generated by nested radicals where the ground field does not necessarely contain the roots of unity to apply Kummer theory. We give a description of the Galois group of extensions over a field K generated by the n-th roots of all elements of K. In chapter 2 we consider field extensions generated by one radical alpha over the ground field Q. We give necessary and sufficient conditions on alpha to generate a field with no other subfields than those generated by a power of alpha. In chapter 3 and 4 we return to the simplification of nested radicals and study nested radicals of special forms. Chapter 5 is an article written together with Alice Gee. We study the Rogers-Ramanujan continued fraction. In 1913 Ramanujan astonished Hardy by giving an identity between a continued fraction and a nested radical. He found this by computing the value of the Rogers-Ramanujan continued fraction at i. In this chapter we give a method to compute nested radicals for singular values of the Rogers-Ramanujan continued fraction.