Radicals in arithmetic

Abstract

Let K be a field. A radical is an element of the algebraic closure of K of which a power is contained in K. In this thesis we develop a method for determining what we call entanglement. This describes unexpected additive relations between radicals, and is encoded in an entanglement group. We give methods for computing the entanglement group, and show how to use these to compute field degrees of radical extensions over the field of rationals. Moreover, we show that these methods give rise to a new explicit method for computing the correction factor in Artin's primitive root conjecture, in a way that more readily admits different generalizations than traditional methods. In chapters 5 and 6 we show how our approach applies to a number of such generalizations of Artin's conjecture. Specifically, we study near-primitive roots, higher rank analogues, and the setting of rank one tori. The last chapter covers an entirely separate topic, and describes an algorithm for enumerating so-called ABC triples. It also reports results from the ABC@home project, a volunteer computing project that has used this algorithm to enumerate all ABC triples up to 10^1