Aims

• To introduce rings and fields and develop their theory.
• To study polynomial rings, and learn why they are of fundamental importance.

• To build on Introduction to Group Theory to study some counting arguments for groups.

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

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. 

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

Group decompositions. Internal and external direct and semidirect products.

Group actions.  Actions and the Sylow theorems, illustrations. 

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 in mid-Summer term.

Elective information

A second course on abstact algebra, assuming a basic knowledge of group theory. 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 much of the material taught in the modules Core Algebra and all of that in Introduction to Group Theory, or are willing to learn the material if necessary.

Prerequisites

• Core Algebra
• Introduction to Group Theory