## MATH3171 Algebra and Numbers

### 10 creditsClass Size: 100

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2005/06

### Pre-requisites

MATH 2080 or MATH2200, or equivalent. (Exclusion: not with MATH2031)

This module is approved as an Elective

### Objectives

describe how algebraic ideas can be generated from number theoretic problems - and how algebra can repay the debt by solving problems in number theory. The main emphasis is on algebraic structure - essentially of rings and fields. By the end of this module students should be able to: (a) Solve various problems in elementary number theory by making use of the Euclidean Algorithm, Fermat?s Little Theorem, Wilson?s theorem, together with properties of unique factorisation in certain number rings. (b) Determine whether or not subsets of rings are subrings, ideals, etc. (c) Be able to construct proofs, similar to those given in the module, which relate to properties of numbers and abstract properties of rings.

### Syllabus

The theory of numbers has long been an attractive topic because of the ease with which one can ask questions: Is every prime of the form 4k + 1 expressible as a sum of two squares ? Why are x = 1 5, y = 3 the only integer solutions of the equation x2 + 2 = y3 ? These and other problems are naturally stated in the language of algebra - and indeed generated many of the ideas of present day algebra. In this course the algebraic structure underlying such problems will be investigated. In particular, we shall look at (number) rings in which elements factorise uniquely into products of primes and at fields with finitely many elements. These latter have real-life applications - to problems in coding theory for example. Topics covered include: 1. Elementary number theory. Existence of infinitely many primes, gcds, Euclidean algorithm, p a prime iff p irreducible, Fundamental Theorem of Arithmetic. Congruences modulo n and their arithmetic. Fermat's and Wilson's theorems. 2. Binary Operations. Definition of 'ring'. Elementary properties derived from axioms. Each finite integral domain a field. Subrings, subfields, ideals. Ideals in Z are principal - similarly for Q[x] (needing Division Algorithm for Q[x]). Similar results for number rings such as Z[{ -2 ]- leading to uniqueness of factorisation into primes and solution of certain Diophantine equations. 3. Factorisation in polynomial rings. Fundamental Theorem of Algebra.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures (20 hours) and examples classes (6 hours).

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

2 hour written examination at end of semester (100%).