Handbook

Being at the intersection of two of the most beautiful and important areas of mathematics, Algebra and Number Theory, this course will build on your abstract algebra background to address number theoretic problems regarding integers or rational numbers, and their generalisation to algebraic numbers.

The course will start with recalling some algebraic background, such as field extensions and basic Galois theory. We will then start developing the theory of algebraic numbers and algebraic integers, ideals in rings of integers and their factorisation, followed by ramification theory which roughly describes the decomposition of prime ideals in a given number field when lifted to finite extensions. We will continue with ideal class groups and units, and if time allows, we will discuss also valuations and several applications.

These concepts will be explored through a series of lectures, along with weekly tutorial exercises, and towards the end of the course, we will also discuss some relevant research topics.