Algebraic number theory can be understood in two correct ways: as algebra applied to number theory, but also as the study of algebraic numbers such as √2 or i∛7.
These numbers occur naturally when one attempts to solve Diophantine equations. For instance, in order to solve the equation y2-2=x3 in integers, it is natural to factor it into (y+√2)(y-√2) = x3. The introduction of the number √2, which is irrational, then raises many questions. For instance, can we say that if y+√2 and y-√2 are coprime, then the equation implies that they are both cubes? And what would coprime mean in this context, exactly?
This would work with regular factorisation of integers into primes, but √2 is of course not an integer, so how can the notion of integers be generalised to include √2? Which of the usual properties of integers will subsist in this new framework?
This module will lay a framework to settle all these questions and more.
Prerequisites
MAU22102 Rings, Fields, and Modules is an important prerequisite for this module, especially the part about field extensions. See the lecture notes on this page, for example, if you would like to go over the material again.