# Modules

## MTH4104 Introduction to Algebra

### Description

This module is an introduction to the basic notions of algebra, such as sets, numbers, matrices, polynomials and permutations. It not only introduces the topics, but shows how they form examples of abstract mathematical structures such as groups, rings and fields, and how algebra can be developed on an axiomatic foundation. Thus, the notions of definition, theorem and proof, example and counterexample are described. The module is an introduction to later modules in algebra.

### Details

Organiser: Dr K Ardakov

Level: 4 Credit value: 15 Semester: B

Overlaps:

• MAS117 Introduction to Algebra

Essential prerequisites:

• MTH4103 Geometry I

Either of

• MTH4107 Introduction to Probability
• MTH4108 Probability I

Assessment: 10% in-term test(s), 90% final exam

Organiser's module website: http://www.maths.qmul.ac.uk/~ardakov/MTH4104

Comment: Assessment was 20% two in-term tests, 80% final exam. Syllabus revised for 2012–13 - HOW? TO BE DONE!

### Syllabus

1. Mathematical basics: proofs, necessary and sufficient conditions, proofs and counterexamples, definitions, existence and uniqueness.
2. Numbers: integers, rationals, real numbers, complex numbers. Induction. Irrationality of √2. Polynomials, matrices.
3. Sets, subsets, functions, relations. One-to-one and onto functions. Equivalence relations and partitions.
4. Division algorithm and Euclidean algorithm. Modular arithmetic. Solving polynomials; remainder and factor theorems.
5. Rings and fields, ideals, factor rings.
6. Groups, subgroups, cyclic groups, Lagrange's Theorem.
7. Permutations, symmetric group, sign.