Courses / Module

RINGS & FIELDS 1

Module code: MT341P
Credits: 5
Semester: 1
Department: MATHEMATICS AND STATISTICS
International:

	Overview
	Module Objective: To give a rigorous introduction to Rings and Fields. Modular Arithmetic, Euclidean algorithm and Euclid's Lemma. Concrete Rings: Polynomial rings and Matrix rings. The abstract definition of a ring. Elementary concepts: subrings, ideals, quotient rings. Divisibility in rings. Prime and irreducible elements. Concrete Fields: the rationals, reals and complex numbers. The integers mod p, p a prime. The abstract definition of a field. Euclidean Domains, Principal Ideal Domains, Unique Factorization Domains. Gaussian Integers and Applications. Gauss� Lemma, Irreducibility of Polynomials.

Overview

Module Objective:
To give a rigorous introduction to Rings and Fields.

Modular Arithmetic, Euclidean algorithm and Euclid's Lemma.
Concrete Rings: Polynomial rings and Matrix rings. The abstract definition of a ring. Elementary concepts: subrings, ideals, quotient rings. Divisibility in rings. Prime and irreducible elements.
Concrete Fields: the rationals, reals and complex numbers. The integers mod p, p a prime. The abstract definition of a field.
Euclidean Domains, Principal Ideal Domains, Unique Factorization Domains. Gaussian Integers and Applications. Gauss� Lemma, Irreducibility of Polynomials.

	Learning Outcomes
	On successful completion of the module, students should be able to: State the axioms defining a ring and a field. Deduce simple properties from these axioms. Provide examples of rings and fields, including noncommutative rings. Determine using the axioms whether a given object is a Principal Ideal Domain or a Unique Factorization Domain. State, prove and apply any definitions and theorems covered in the course. Determine whether a given polynomial is irreducible. Determine ring theoretic properties of concrete commutative rings. Use the methods from the course to solve problems.

Learning Outcomes

Teaching & Learning methods

Assessment

	Autumn Supplementals/Resits
	Are supplemental/resit registrations permitted? Yes Continuous assessment mark is carried forward to the Autumn. University scheduled written examination (Autumn): 120 minutes, On-campus

	Pre-Requisites
	MT241P and MT242P. Equivalent modules would also suffice.

Timetable

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Courses / Module

RINGS & FIELDS 1

Semester 1

Semester 2