## MATH2027 Rings and Polynomials

### 10 creditsClass Size: 90

Module manager: Professor Karin Baur
Email: k.u.baur@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2022/23

### Pre-requisite qualifications

MATH2020 or MATH2022

### This module is mutually exclusive with

 MATH2025 Algebraic Structures 2 MATH2026 Rings, Fields and Polynomials

Module replaces

MATH2026

This module is not approved as a discovery module

### Module summary

Given two integers, we may add, subtract or multiply them to produce new integers. The same is true for two polynomials, or two matrices. This leads to the notion of a ring: an algebraic system in which addition, subtraction and multiplication may be performed, satisfying standard properties. By studying rings, we can obtain results which apply simultaneously to all examples of the above kind, as well as others. For example, in arithmetic modulo a positive integer n, we specify that two integers are congruent modulo n if they have the same remainder on dividing by n. This gives rise to a ring with finitely many elements (the n possible remainders), under addition and multiplication mod n. Another classical example is the Gaussian integers: complex numbers which have integer real and imaginary parts. Rings are one of the fundamental languages of mathematics, and they play a key role in many areas, including algebraic geometry, number theory, Galois theory and representation theory. The aim of this module is to give an introduction to rings. The emphasis will be on interesting examples of rings and their properties. Since a ring can be considered as a generalisation of the integers, it is natural to ask whether properties of the integers hold for other rings. For example, the Fundamental Theorem of Arithmetic states that every positive integer can be written in a unique way as a product of prime numbers. We shall see that a corresponding theorem also holds in other rings, such as polynomials with real coefficients.

### Objectives

On completion of this module, students should be able to:

a) accurately reproduce appropriate definitions;
b) state the basic results about rings and fields, and reproduce short proofs;
c) identify subrings, ideals and units in the main examples of rings;
d) use the First Isomorphism Theorem to exhibit isomorphisms between rings;
e) demonstrate understanding of unique and non-unique factorisation;
f) use standard tests to determine the irreducibility of polynomials.

### Syllabus

1. Definitions, basic properties and examples of rings, commutative rings, subrings;
2. Homomorphisms and isomorphisms, ideals, principal ideals, factor rings and the First Isomorphism Theorem;
3. Integral domains, units, polynomial rings;
4. Euclidean domains and examples, principal ideal domains and examples;
5. Greatest common divisors, associates, prime elements, irreducible elements;
6. Unique factorisation domains and examples;
7. The rings obtained by adding a square root of an integer to the integers, and their properties; fields as factor rings of principal ideal domains;
8. Tests for irreducibility of polynomials: the existence of roots, the Rational Root Test, Eisenstein's Criterion and Gauss's Lemma;
9. Fields, the characteristic of a field, a field is a vector space over a subfield.

### Teaching methods

 Delivery type Number Length hours Student hours Workshop 10 1.00 10.00 Lecture 22 1.00 22.00 Private study hours 68.00 Total Contact hours 32.00 Total hours (100hr per 10 credits) 100.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Written, assessed work throughout the semester with feedback to students.

### Methods of assessment

Coursework
 Assessment type Notes % of formal assessment Written Work * 15.00 Total percentage (Assessment Coursework) 15.00

There is no resit available for the coursework component of this module. If the module is failed, the coursework mark will be carried forward and added to the resit exam mark with the same weighting as listed above.

Exams
 Exam type Exam duration % of formal assessment Open Book exam 2 hr 85.00 Total percentage (Assessment Exams) 85.00

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