MAU34109 - Algebraic number theory
Course information
Lecturer:
Nicolas Mascot
Syllabus
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.
MAU23101 Intoduction to Number Theory is also useful, especially the section about congruences, but not as crucial as MAU22102.
Assessment
10% homework assignments + 90% final exam.
Lecture notes
Lecture notes for this module are available here.
Homework
Homework assignments (and solutions) will be posted here later.
You must submit your homework assignments before the deadline, by bringing them to the lecture or by emailing them to the T.A.. If you choose the latter option, you must typeset your answers with LaTeX.
The solutions will be made available immediately after the deadline, so late submissions will not be accepted.
Review material for the exam
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