## MATH2026 Rings, Fields and Polynomials

### 10 creditsClass Size: 150

Module manager: Professor R Marsh
Email: R.J.Marsh@leeds.ac.uk

Taught: Semester 2 View Timetable

Year running 2015/16

### Pre-requisite qualifications

MATH2020 or MATH2022

### This module is mutually exclusive with

 MATH2025 Algebraic Structures 2 MATH2033 Rings, Polynomials and Fields

Module replaces

MATH2025

This module is approved as a discovery module

### Module summary

A ring is an algebraic system in which addition, subtraction and multiplication may be performed. Integers, polynomials and matrices all provide examples of rings, as do the rational, real and complex numbers, so this notion generalizes an important range of mathematical structures. They are studied in this module. One topic studied is the generalization to some other rings of the Fundamental Theorem of Arithmetic, that every positive integer can be written in a unique way as a product of primes. This holds for polynomials in a single variable over the rational numbers, but not over the integers. Another topic is Kronecker's Theorem, showing that every non-constant polynomial with coefficients in a field has a root in some possibly larger field. For example, every such polynomial over the rational numbers has a root in the complex numbers. These techniques also allow one to study finite fields, which play a fundamental role in modern communications.

### 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,
and compute minimal polynomials of algebraic elements in extension fields.
g) Determine the minimal polynomial of a matrix.

### Syllabus

1. Definitions, basic properties and examples of rings; subrings; homomorphisms and isomorphisms, ideals, factor rings and the First Isomorphism Theorem; units; integral domains, fields and the characteristic of a field.
2. Principal Ideal Domains and examples; greatest common divisors; irreducible elements; unique factorization property; the factor ring of a Principal Ideal Domain by an irreducible element is a field.
3. Tests for irreducibility of polynomials using the existence of roots, the Rational Root Test and Eisenstein's Criterion; Gauss's Lemma.
4. Algebraic elements in field extensions and minimal polynomials; statement of Cayley-Hamilton theorem and existence
of the minimal polynomial of a matrix; Kronecker's Theorem; finite fields, their construction via factor rings of polynomial rings, and their number of elements.

### 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

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

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 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