Monogenic number fields

Abstract

Determining whether the ring of integers OK of an algebraic number field K of degree n admits a power integral basis is one of the classic problems in algebraic number theory. In other words, we want to determine whether there exists α ∈ OK such that {1, α, . . . , αn−1} is a Q-basis for K. This question dates back to the 1960s and was introduced by a German mathematician, Helmut Hasse. In this thesis, we will study the monogenicity of cubic number fields and their lift to monogenic sextic number fields. After recalling some background material on algebraic number theory and related topics, we will focus on specific cubic fields such as pure cubic fields and cyclic cubic fields. Next, we will study the lifting of all monogenic cyclic cubic fields to monogenic sextic fields. This thesis was supported by Bo˘gazi¸ci University Research Fund Grant Number 19082.