# Algebra 2

## Unit aims

To develop the theory of commutative rings, and to apply it to solving problems concerning the factorisation of polynomials, algebraic numbers, ruler-and-compass constructions, and the construction of roots of polynomials.

## Unit description

Algebraic structures -- such as groups, rings, and fields -- are pervasive in mathematics. This course focuses on (commutative) rings, which are sets equipped with two (commutative) operations (called addition and multiplication), and that contain an additive identity and an additive inverse for each element of the set. A fundamental example of a ring is **Z**, the set of integers; other important examples include **Q**, **Z** modulo n, and **Q**[X], which is the set of polynomials in X with rational coefficients. A fruitful way to study rings and their properties is to study "homomorphisms" between rings: a homomorphism is a map that preserves addition and multiplication (just as a linear transformation preserves vector addition and scalar multiplication). Using homomorphisms and generalised modular arithmetic, we develop means of determining when a ring has additional nice properties, such as having multiplicative inverses for each nonzero element of the ring. This is a very beautiful and clean theory; in proving the theorems, the students will learn some new techniques and strengthen their proof-writing skills.

## Relation to other units

This unit has some relationship to (but is independent of), Linear Algebra 2 and Representation Theory and has a stronger relationship to Algebraic Number Theory. It provides the foundation for Galois Theory.

## Learning objectives

After taking this unit, students should be able to state the basic definitions and results in the subject, to utilise the fundamental proof techniques, and to solve problems similar to those worked in the lectures and set as homework.

## Transferable skills

The ability to understand and apply general theory, and the acquisition of facility in calculating in a variety of number-systems.

## Syllabus

Rings and subrings.

Homomorphisms, ideals, and quotient rings.

Basic homomorphism theorems.

Integral domains and fields.

Euclidean domains, principal ideal domains, and unique factorisation domains

Gauss' Lemma and consequences.

Testing polynomials for irreducibility.

Field extensions and algebraic elements.

The characteristic of a field and finite fields.

Application to ruler and compass constructions

## Reading and References

No particular text will be used, but there are many books on abstract algebra in the university library. These include:

*A Book of Abstract Algebra by *Charles C. Pinter* (free on archive.org)*

*Rings, fields, and groups* by R.B.J.T. Allenby

*Contemporary abstract algebra* by Joseph A. Gallian

*A first course in abstract algebra* by John B. Fraleigh

*Abstract algebra: a first course* by Larry Joel Goldstein

**Unit code:** MATH21800

**Level of study:** I/5

**Credit points:** 20

**Teaching block (weeks):** 2 (13-24)

**Lecturer:** Professor Tim Dokchitser

## Pre-requisites

Foundations and Proof, Introduction to Group Theory, Linear Algebra and Geometry

## Co-requisites

None

## Methods of teaching

There are 3 lecture classes and 1 problems/feedback session each week. The course is based on the lectures and exercises. The basic lecture notes will be posted and solutions to most of the excerises will be distributed.

The last 2 weeks of the course are devoted to review and revision, and in this time exercises (both assigned and not assigned) and previous exam questions will be addressed.

Besides the problems classes, there is also a weekly office hour during which students can ask questions about lectures and execerises.

## Methods of Assessment

The pass mark for this unit is 40.

The final mark is calculated as follows:

- 85% from a 2 hour 30 minute exam in May/June
- 15% from assigned homework questions

NOTE: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.