Synopsis for Introduction to Fields

Overview

Informally, finite fields are generalisations of systems of real numbers such as the rational or the real numbers— systems in which the usual rules of arithmetic (including those for division) apply. Formally, fields are commutative rings with unity in which division by non-zero elements is always possible. It is a remarkable fact that the finite fields may be completely classified. Furthermore, they have classical applications in number theory, algebra, geometry, combinatorics, and coding theory, and they have newer applications in other areas. The aim of this course is to show how their structure may be elucidated, and to present the main theorems about them that lead to their various applications.

Learning Outcomes

Students will have a sound knowledge of field theory including the classification of finite fields. They will have an appreciation of the applications of this theory.

Synopsis

Fields, characteristic of a field, field extensions, algebraic and transcendental elements. The degree of a field extension and the tower theorem. Constructions with ruler and compass. Symbolic adjunction of roots. Multiple roots. Finite fields: existence, uniqueness, primitive element, number of irreducible polynomials over the field with elements, subfields of finite fields.

1. P.J. Cameron, Introduction to Algebra (2nd. ed., OUP, 2008) pp. 99-103, 220-223, 268-276.
2. Joseph J. Rotman, A First Course in Abstract Algebra (Second edition, Prentice Hall, 2000), ISBN 0-13-011584-3. Chapters 1,3.

Further Reading

1. Michael Artin, Algebra (2nd ed. Pearson, 2010) Chapter 13
2. I. N. Herstein, Topics in Algebra (Wiley, 1975). ISBN 0-471-02371-X 5.1, 5.3, 7.1. [Harder than some, but an excellent classic. Widely available in Oxford libraries; still in print.]
3. P. M. Cohn, Classic Algebra (Wiley, 2000), ISBN 0-471-87732-8, parts of Chapter 6. [This is the third edition of his book on abstract algebra, in Oxford libraries.]