# Abstract Algebra 1

## Overview

• Credit value: 15 credits at Level 6
• Convenor: Dan Mcveagh
• Assessment: coursework (20%) and a two-hour examination (80%)

## Module description

In this module you will learn key concepts from abstract algebra. Combined with the module Abstract Algebra 2, the module will cover material on algebraic structures such as groups, rings, fields and vector spaces, giving you a thorough grounding in these topics.

The module is studied via distance learning. Each week, you will be guided through a series of learning steps. These include short instructional videos, online quizzes to test your understanding, livestreamed face-to-face examples classes, full course notes, and exercises to try at home.

### Indicative syllabus

• Revision of key prerequisite material on sets, functions and matrices
• Binary relations and equivalence relations
• Binary operations - definition, properties and examples
• Definition of a group - first examples and properties
• Subgroups, the subgroup test, further examples
• Cosets and Lagrange’s Theorem
• Linear congruences and the integers modulo n
• Fields: definition and examples of finite and infinite fields
• Definition and properties of the characteristic of a field, and general results on fields
• Vector spaces over arbitrary fields: formal axiomatic definition, properties, examples and results

## Learning objectives

By the end of this module, you will:

• have knowledge and understanding of, and the ability to use, mathematical methods and techniques
• have knowledge and understanding of a range of results in mathematics
• appreciate the need for proof in mathematics, and be able to follow and construct mathematical arguments
• understand the importance of assumptions and have an awareness of where they are used and the possible consequences of their violation
• appreciate the power of generalisation and abstraction in the development of mathematical theories
• have a deeper knowledge of particular areas of mathematics.