| Unit name | Algebraic Number Theory |
|---|---|
| Unit code | MATH31110 |
| Credit points | 10 |
| Level of study | H/6 |
| Teaching block(s) |
Teaching Block 2 (weeks 13 - 24) |
| Unit director | Dr. Jordan |
| Open unit status | Not open |
| Pre-requisites |
Level 1 Pure Maths or Number Theory & Group Theory |
| Co-requisites |
None |
| School/department | School of Mathematics |
| Faculty | Faculty of Science |
The theme for this unit is the use of algebra to solve certain problems in number theory, and also the way that some problems in number theory have stimulated the development of certain branches of algebra. The flavour of the course is algebraic, using polynomials, field extensions, rings, ideals, etc. together with some linear algebra and some basic facts from first year group theory. The problems in number theory that shall be dealt with include the following: the Two Squares Theorem, the solution of various Diophantine equations, and some cases of Fermat's Last Theorem.
Aims
The course will give a brief introduction to some ways in which results from algebra can be used to solve problems in number theory.
The central object of study will be the ring of integers of a quadratic number field. In this setting there will be a great emphasis on determining when unique factorisation holds or fails. Methods will be developed for solving various special types of Diophantine equation, culminating with a proof of Fermat's Last Theorem for cubes.
Problem-solving will be a major part of the course.
Syllabus
Preliminaries: standard notation; revision of the integers mod(n); definitions and simple properties of field, integral domain, ring (2 lectures)
Subrings and subfields of the complex numbers: test for a subset of C to be a subring or a subfield; every subring of C contains Z and every subfield contains Q; examples involving the square-roots of square-free integers; the Gaussian integers and Gaussian numbers; brief discussion of field extensions and degree; characterisation of the quadratic extensions of Q (2 lectures)
Euclidean domains and unique factorisation domains: definitions and basic theory of Euclidean domains, units, irreducible elements, unique factorisation domains; use of arithmetic language such as "relatively prime" in arbitrary integral domains; every Euclidean domain is a U.F.D.; relatively-prime factors of perfect nth powers in a U.F.D. (2 lectures)
Quadratic integers: definition of the ring of integers of the quadratic number field corresponding to a square-free integer d and form of the elements according as d is not or is congruent to 1 mod(4); definition and basic properties in this context of the norm; characterisation of units by their norms; identification of the units for all negative square-free d; identification of the units when d = 2; use of the norm to solve certain quadratic Diophantine equations; unsolvability of some such equations; use of the norm to identify irreducible elements; examples which are not U.F.D.'s; proof of the Euclidean property when d = -11, -7, -3, -2, -1, 2, 3, 5, 13; discussion of the case d = -3; identification of those negative d not congruent to 1 mod(4) for which unique factorisation holds (5 lectures)
Some Diophantine equations: definition and basic properties of associate elements; solution of some Diophantine equations of the type X2 + 2 = Y3, (3 lectures)
Sums of two squares: Euler's Criterion; the 2-squares theorem; characterisation of the irreducible Gaussian integers (1 lecture)
Proof of Fermat's Last Theorem for cubes (2 lectures)
Relation to Other Units
Other approaches to number theory are given in the Level 3 Number Theory unit.
After taking this unit, students should be able to state the basic definitions and results concerning the ring of integers of a quadratic number field; they should also be able to apply these ideas to solve standard types of number-theoretic problems (examples of which will be given either as worked examples in the notes or as exercises with solutions provided later). The students will need to be able to do simple arithmetic calculations quickly and accurately, with the help of a calculator if necessary. Calculators will be permitted in the exam. Written information will be provided towards the end of the unit about which parts are examinable.
Transferable Skills:
The ability to apply abstract ideas to solve specific types of number-theoretic problems, and the ability to produce clear logical arguments.
The course will be taught by lectures based on duplicated typed notes. Exercise sheets and model solutions will also be provided.
The assessment mark for Algebraic Number Theory is calculated from a 1½-hour written examination in May/June consisting of THREE questions. The mark will be determined from the best TWO answers. Calculators of the approved type (non-programmable, no text facility) may be used.
No particular book will be used, but guidance will be given during the course for further or alternative reading.
