MATH20212 - 2011/2012
Unit specification
Aims
The programme unit aims to introduce the algebraic structures of rings and fields; describe the quotient structure and its connection with homomorphisms of rings; present important examples rings and develop some of their properties with particular emphasis on polynomial rings and factorisation in rings.
Brief description
This course builds on Algebraic Structures 1, which is a prerequisite, and continues the strong emphasis on examples.
The algebraic structures of rings and fields will be introduced. The construction of quotient rings and the relationship with homomorphisms is one of the main themes. These ideas will be used to construct roots of polynomials in extension fields. Factorisation in polynomial rings and rings of integers of number fields will also be studied.
Intended learning outcomes
On completion of this unit successful students will be able to:
- demonstrate their knowledge of the definition of a ring and important examples;
- demonstrate their understanding of the quotient construction in theoretical terms and also in particular contexts;
- demonstrate their understanding of how to produce roots of polynomials in extension fields and be able to compute in such fields;
- solve a range of problems which require understanding of rings and fields;
- apply theoretical results to computations in particular examples of rings.
Future topics requiring this course unit
Most, possibly all, algebra courses in years 3 and 4.
Syllabus
- Definitions and examples of rings (rings of numbers, rings of matrices, quaternions, rings of endomorphisms, group rings, rings of polynomials, subrings); [4 lectures]
- Domains, fields and division rings; nilpotent and idempotent elements, products of rings; (many) examples; with students gaining familiarity with the ideas and examples through attempting exercises. [4]
- Isomorphisms and homomorphisms (of rings): what is preserved and reflected; kernel of a homomorphism, ideals; principal ideals, operations on ideals. [4]
- The quotient construction (for rings): the construction and connection with homomorphisms; maximal ideals; ideals of the quotient ring; examples. [3]
- Polynomial rings and unique factorisation: polynomial rings; division algorithm; unique factorisation. [3]
- Constructing roots of polynomials: construction of extension fields; examples, including finite fields. [4]
Textbooks
- J.B. Fraleigh, A First Course in Abstract Algebra, (any edition: the library has many copies) Addison-Wesley (recommended but not essential).
Also similar books like:
- R.B.J.T. Allenby, Rings, Fields and Groups: an Introduction to Abstract Algebra, Addison-Wesley.
Learning and teaching processes
Two lectures and one examples class each week. In addition students should expect to do at least four hours private study each week for this course unit.
Assessment
- Coursework; Weighting within unit 20%
- 2 hours end of semester examination; Weighting within unit 80%