Content: Algebraic number theory is the study of algebraic numbers, which are the roots of monic polynomials
with rational coefficients, and algebraic integers, which are the roots of monic polynomials with integer coefficients. So, for example, the nth roots of natural numbers are algebraic integers, and so is
The study of these types of numbers leads to results about the ordinary integers, such as determining which of them can be expressed as the sum of two integral squares, proving that any natural number is a sum of four squares and, as a much more advanced application, which combines algebraic number theory with techniques from analysis, the proof of Fermat's Last Theorem.
One of the differences between rings of algebraic integers and the ordinary integers, is that we do not always get unique factorisation into irreducible elements. For example, in the ring
it turns out that 6 has two distinct factorisations into irreducibles:
However, we do get a unique factorisation theorem for ideals, and this is the central result of the module.
This main result will be followed by some more straightforward geometric material on lattices in , with applications to sums of squares theorems, and then finally various groups associated with the ideals in a number field.
- Algebraic numbers, algebraic integers, algebraic number fields, integral bases, discriminants, norms and traces.
- Quadratic and cyclotomic fields.
- Factorisation of algebraic integers into irreducibles, Euclidean and principal ideal domains.
- Ideals, and the prime factorisation of ideals.
- Minkowski's Theorem.
- The ideal class group.
- Units; the unit group of a quadratic field.
- Using the class group and unit group to solve Mordell and Pell equations in rational integers.
- To demonstrate that uniqueness of factorisation into irreducibles can fail in rings of algebraic integers, but that it can be replaced by the uniqueness of factorisation into prime ideals.
- To apply the techniques in the course to solve Mordell and Pell equations.
Objectives: By the end of the course students will:
- Be able to compute norms and discriminants and to use them to determine the integer rings in algebraic number fields;
- Be able to factorise ideals into prime ideals in algebraic number fields in straightforward examples;
- Understand the use of the class group and unit group to solve Mordell and Pell equations in rational integers.
This module is based on the book Algebraic Number Theory and Fermat's Last Theorem, by I.N. Stewart and D.O. Tall, published by A.K. Peters (2001). The contents of the module forms a proper subset of the material in that book. (The earlier edition, published under the title Algebraic Number Theory, is also suitable.)
For alternative viewpoints, students may also like to consult the books A Brief Guide to Algebraic Number Theory, by H.P.F. Swinnerton-Dyer (LMS Student Texts # 50, CUP), or Algebraic Number Theory, by A. Fröhlich and M.J. Taylor (CUP).