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MAU34109 Algebraic number theory

Module Code	MAU34109
Module Title	Algebraic number theory
Semester taught	Semester 1
ECTS Credits	5
Module Lecturer	Prof. Nicolas Mascot
Module Prerequisites	MAU22102 Fields, rings and modules and MAU23101 Introduction to number theory

Assessment Details

This module is examined in a 2-hour examination at the end of Semester 1.
Continuous assessment contributes 20% towards the overall mark.
The module is passed if the overall mark for the module is 40% or more. If the overall mark for the module is less than 40% and there is no possibility of compensation, the module will be reassessed as follows:
1) A failed exam in combination with passed continuous assessment will be reassessed by an exam in the supplemental session;
2) The combination of a failed exam and failed continuous assessment is reassessed by the supplemental exam;
3) A failed continuous assessment in combination with a passed exam will be reassessed by one or more summer assignments in advance of the supplemental session.

Contact Hours

11 weeks of teaching with 3 lectures per week.

Learning Outcomes

On successful completion of this module, students will be able to

Determine the main invariants of a number field, such as its ring of integers, discriminant, class group, and fundamental units.
Apply these tools to solve Diophantine equations, e.g. to discover that y² = x³ -292 admits the solutions (98, ±970) and that these are the only integer solutions.

Module Content

Number fields: Review of field extensions, degrees, and minimal polynomial; trace and norm, resultant.
Algebraic integers: Integral elements, orders, discriminants.
Ideals and factorisation: (Non-)uniqueness of factorisation, prime and maximal ideals, finite fields, Dedekind domains, factorisation of ideals, ramification.
The class group: Failure of unique factorisation and remedy, finiteness of the class group, application to Diophantine equations.
Units: Roots of unity, Dirichlet's theorem, application to the Pell-Fermat equation and to class group computations.
Geometry of numbers (if time permits): Euclidean lattices, Minkowski's theorem, proof of Minkowski's bound and of Dirichlet's theorem.

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 over the 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?