## MATH3171 Algebra and Numbers

### 10 creditsClass Size: 100

Module manager: Professor D. Macpherson
Email: H.D.Macpherson@leeds.ac.uk

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2008/09

### Pre-requisite qualifications

MATH2080 or MATH2200, or equivalent.

### This module is mutually exclusive with

 MATH2032 Rings, Polynomials and Fields MATH3044 Number Theory

This module is approved as an Elective

### Module summary

The theory of numbers has long been an attractive topic because of the ease with which one can ask questions: Which primes can be expressed as a sum of two squares? Why do certain polynomial equations in two variables have only a small number of integer solutions? 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.

### 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

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

 Delivery type Number Length hours Student hours Example Class 11 1.00 11.00 Lecture 20 1.00 20.00 Private study hours 69.00 Total Contact hours 31.00 Total hours (100hr per 10 credits) 100.00

### Methods of assessment

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

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated