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MA3A6 Algebraic Number Theory

Lecturer: Simon Myerson

Term(s): Term 1

Status for Mathematics students: List A

Commitment: 30 one-hour lectures.

Assessment: Three-hour examination (85%), assignments (15%)

Formal registration prerequisites: None

Assumed knowledge: It is an extremely good idea to revise finitely generated abelian groups (Algebra I) and ideals (Algebra II) before starting this course. MA251 Algebra I: Advanced Linear Algebra - Finitely generated abelian groups, especially: generating sets, free basis, change of basis, classification of Finitely Generated Abelian Groups, subgroups of free abelian groups. MA249 Algebra II: Groups and Rings - Rings, fields, ideals, factorisation of polynomials, Gauss' Lemma, Eistenstein's Criterion, the First Isomorphism Theorem for rings, the Third Isomorphism Theorem for groups, Domains (Integral Domains, UFDs, PIDs). MA132 Foundations - this course is always assumed knowledge! But modular arithmetic and solving congruences in the integers are particularly relevant here.

Useful background: MA257 Introduction to Number Theory - Prime factorisations in the integers. Solving congruences in integers. The Gaussian integers, potentially Minkowski's theorem on lattices if we have time. MA249 Algebra II: Groups and Rings as well as the assumed knowledge, you might have seen an example of an integral domain that's not a UFD; that's nice motivation for this course. The Chinese Remainder Theorem might also come up.

Synergies: MA3D5 Galois Theory - Like this course, Galois Theory studies algebraic numbers; if you're interested in one you will probably enjoy the other as well. Some results will be stated without proof in this course, and proved in Galois Theory. The two courses have a lot of overlap in the pre-requisites, especially around polynomial rings and factorising integer polynomials.

Content: Algebraic number theory is the study of algebraic numbers, which are the roots of monic polynomials

 x^n+a_{n-1}x^{n-1} \cdots+a_1x+a_0

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

 \frac{{\sqrt {5}} + 1}{2}

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 factorization into irreducible elements. For example, in the ring

 \{a+b\sqrt{-5} \mid a,b \in Z \} ,

it turns out that 6 has two distinct factorizations into irreducibles:

 6=2\times 3

and

 6=(1-{\sqrt{-5}})\times(1+{\sqrt{-5}}) .

However, we do get a unique factorization 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  \R^n , 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.
  • Factorization of algebraic integers into irreducibles, Euclidean and principal ideal domains.
  • Ideals, and the prime factorization of ideals.
  • Lattices.
  • Minkowski's Theorem. Application: every integer is the sum of four squares.
  • The geometric representation of algebraic numbers.
  • The ideal class group.

Aims: To demonstrate that uniqueness of factorization into irreducibles can fail in rings of algebraic integers, but that it can be replaced by the uniqueness of factorization into prime ideals.

To introduce some geometric lattice-theoretic techniques and their applications to algebraic number theory.

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 algabraic number fields;
  • be able to factorize ideals into prime ideals in algebraic number fields in straightforward examples;
  • understand the proof of Minkowski's Theorem on lattices, and be able to apply it, for example, to prove that all positive integers are the sum of four squares.

Books:

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).

Additional Resources

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