MA2ALANU-Algebra

Module Provider: Mathematics and Statistics
Number of credits: 20 [10 ECTS credits]
Level:5
Semesters in which taught: Semester 1 module
Pre-requisites: MA1LANU Linear Algebra and MA0FMNU Foundations of Mathematics
Non-modular pre-requisites:
Co-requisites:
Modules excluded:
Current from: 2022/3

Module Convenor: Dr Basil Corbas
Email: b.corbas@reading.ac.uk

Type of module:

Summary module description:

This module is an introduction to the basic concepts of algebra, centred around group, ring and field theory.

The Module lead at NUIST is Dr Raul Sanchez GalanÌý(raul.galan.13@alumni.ucl.ac.uk).

Aims:

To develop the basic theory of groups, rings and fields; to illustrate the fascinating and unexpected interconnections among seemingly unrelated topics, especially between concrete and "abstract" algebra.

Assessable learning outcomes:

By the end of the module students are expected to be able to:

1. Work with groups, subgroups and quotient groups;


2. Recognise homomorphisms and establish simple isomorphisms;


3. Work with permutations expressed in cycle notation;


4. Recognise subrings and ideals;


5. Construct quotient rings;


6. Construct simple algebraic extensions.

Additional outcomes:

By the end of the course, students are expected to have acquired skill in logical reasoning and construction of proofs.

Outline content:

The first half of the module studies in detail the basic theory of groups, i.e.. sets equipped with an abstract operation of multiplication satisfying certain axioms modelled on a plethora of motivating examples. This provides both an understanding of the common properties of many different kinds of mathematical objects and insight into the differences between them. In particular the following topics will be discussed:

• Groups, subgroups, quotient groups, Lagrange's Theorem, cyclic groups, symmetric groups, homomorphisms and isomorphisms, Cayley's Theorem.


• The second part of the module proceeds along the same pattern to introduce the theory of rings and fields. In particular the following topics will be discussed:


• Rings, subrings, ideals, the quotient ring with respect to an ideal, ring homomorphisms, polynomials and polynomial rings, algebraic and transcendental extensions, finite fields.

Brief description of teaching and learning methods:

Lectures supported by tutorials and problem sheets.