# Groups Rings and Fields

Course category:
2nd year
Module code:
MAT00005I
Year:
2011/12
Term:
Spring
Term:
Summer
Credits:
20

## Aims

• To develop the theory of groups, and introduce the more complicated algebraic structure of rings and fields.
• To introduce polynomial rings, and learn why they are of fundamental importance.

## Learning Outcomes

• Be comfortable with the axiomatic definitions of groups, rings, and fields, and be familiar with common examples.
• Be able to deduce simple properties from the axioms.
• Understand the concept of groups acting on sets.
• Appreciate the Sylow Theorems.
• Have a good understanding of homomorphisms of groups and rings, and to be able to find the kernel and image of a given homomorphism.
• Be familiar with the definitions of integral domain, principal ideal domain, unique factorisation domain and Euclidean ring, and to know the common examples.
• Appreciate the role of polynomial rings in constructing finite fields.

## Syllabus

Group actions.  Actions and the Sylow theorems, illustrations.

Rings.  Definition of rings and examples (including matrix rings and rings of functions), consequences of axioms, subrings.

Fundamental theorem of ring homomorphisms.  Homomorphisms, ideals, quotient rings, kernels and homomorphism theorems.

Special rings.  Integral domains, principal ideal domains and fields.

Ideals.  Prime and maximal ideals: for commutative R, R/I is an integral domain if and only if I is prime; R/I is a field if and only if I is maximal.

Division in commutative rings with identity; unique factorization domains; Euclidean rings.  Examples including polynomial rings. Euclidean rings are PIDs, PIDs are UFDs.

Division.  Further examples.

Extension fields.  Splitting fields of polynomials; constructing finite fields; the group of units of a finite field is cyclic.

## Recommended texts

J B Fraleigh, A First Course Abstract Algebra, Addison-Wesley.

## Teaching

• 3 lectures and 1 seminar per week in Weeks 2-10 of the Spring term and Weeks 1-3 of the Summer term
• 2 revision classes Week 4 of the Summer term.

## Assessment

Three hour closed examination.

## Elective information

A second course on abstact algebra, assuming a basic knowledge of group thoery. Please check prerequisites carefully before asking to take this module as an elective. In choosing this module as an elective it will be assumed that you are familiar with all the material taught in the first year courses Calculus and Core Algebra, or are willing to learn the material if necessary.