1.  Introduction:  This unit is one of the main second year units in Pure Mathematics. It leads on from Algebra I which is a prerequisite and continues with this introduction to contemporary algebra and axiomatic methods in mathematics. Its main topic is Ring Theory, a very powerful tool of mathematics. Apart from the general notions of rings, ideals and homomorphisms, the course deals with the theory of polynomials, an introduction to field theory and the theory of factorization. This unit is very important as it is a prerequisite for most third year Pure Mathematics courses: Ring Theory, Analytic Number Theory and Number theory, Set Theory and Logic, Theory of Finite Groups, Representation Theory, Galois Theory, and the Pure Mathematics Reading Course.

2.  Hours, Credit, Assessment:  The course is a 10 UCU unit of 20 lectures and three or four support classes. Assessment is by examination (80%) and coursework (20%) via assessed home work.

3.  Overview:  The course is the second step towards an introduction to axiomatic systems. (The first was the development of the notion of a group in Algebra I.) The idea of developing a theory from axioms is one of the central methods of contemporary mathematics. This requires developing new skills and, especially, the skill of working confidently in an axiomatic environment. The course starts by introducing the notions of ring, subring, ideal, homomorphism, isomorphism, quotient ring, and showing a variety of examples. After introducing the notion of a maximal ideal and a field, it becomes possible to show a powerful method of constructing fields as quotients of commutative rings modulo maximal ideals. At this stage there will also be an introduction to finite fields. This is a starting point of commutative ring theory. The notions of principal ideal and divisibility are steps to higher arithmetics with divisibility theory, and to the theory of polynomial rings. The division algorithm opens a door to constructing non-prime finite fields by giving techniques for computations in finite fields. The next portion of abstract theory is the theory of principal ideal domains and the theory of Euclidean domains. The theorem on unique factorization in Euclidean rings is the principal result. Complementary to it is the unique factorization theorem for polynomial rings over a unique factorization ring. The course ends with results on factorization of polynomials with integer coefficients.

4.  Recommended Reading:  All of the following should be in the Library and there are many other books in the library which covers the material of the course.

1. J.B. Fraleigh. A first course in abstract algebra;

2. R.A. Dean. Elements of abstract algebra;

3. I.N. Herstein. Topics in Algebra.

4. B. Hartley and T. Hawkes, Rings, Modules and Linear algebra.

In addition, lecture notes may be provided to students and made available on the UEA intranet.

Main definitions and elementary properties: Subrings, commutative rings, units, fields, integral domains, simple rings, ideals, principal ideals. Matrix rings. (4 lectures)

Homomorphism and isomorphism, quotient ring, quotient ring of a maximal ideal, finite fields. (2 lectures)

Polynomial rings: evaluations, roots, division algorithm, remainder theorem, irreducible polynomials, coprime polynomials. (2 lectures)

Maximal ideals and fields. Construction of finite fields of non-prime order. Computations in finite fields. (3 lectures)

Principal ideal domains and Euclidean domains. Factorization. (2 lectures)

Unique factorization domains. Theorem saying that R[x] is UFD if R is a UFD. (2 lectures)

Polynomials with integer coefficients, content function, Gauss' lemma and Eisenstein's irreducibility criterion. (3 lectures)

The ring of algebraic integers, possibly Dedekind domains. (2 lectures)