Module Identifier | MA32310 | ||

Module Title | RINGS AND FIELDS | ||

Academic Year | 2000/2001 | ||

Co-ordinator | Dr V C Mavron | ||

Semester | Semester 1 | ||

Other staff | Dr C R Fletcher | ||

Pre-Requisite | MA20410 , MA21410 | ||

Course delivery | Lecture | 19 x 1hour lectures | |

Seminars / Tutorials | 3 x 1hour example classes | ||

Assessment | Exam | 2 Hours (written examination) | 100% |

Resit assessment | 2 Hours (written examination) | 100% |

**General description**

Rings are abstract algebraic structures which exhibit some of the common algebraic properties of the integers and of plynomials. The general theory of such structures is developed. Fields, which are particular types of ring, have an especially rich algebraic structure. Important basic classes of rings and fields are contructed and examined. Field theory provides powerful techniques for solving a wide range of mathematical problems which include classical problems of Greek geometry, such as angle trisection and circle squaring, and the later question of the solvability of quintic equations.

**Aims**

To investigate the algebraic properties of rings, and in particular to examine common concepts of the integers and of polynomials in this general setting. To highlight the special algebraic structure of fields and to construct examples of fields. To apply field theory to the classical problems of geometry.

**Learning outcomes**

On completion of this module, students should be able to:

- verify that certain algebraic systems are rings or integral domains or fields;
- construct the quotient ring with respect to an ideal;
- construct ring homomorphisms and use the isomorphism theorem;
- factorise polynomials over number systems into irreducibles;
- construct simple extension fields using irreducible polynomials;
- construct finite fields;
- compute the degrees of finite extensions using intermediate fields or minimum polynomials of generators;
- solve the classical problems of geometry concerning ruler and compass constructions using field theory.

**Syllabus**

1. Rings, commutative rings with identity, integral domains, fields.

2. Ideal (prime, maximal, principal), quotient ring, ring homomorphism and the isomorphism theorem.

3. Euclidean domains, principal ideal domains.

4. Divisibility and the construction of the greatest common divisor of two elements in a Euclidean domain.

5. Quotient fields.

6. Factors and roots of polynomials.

7. Irreducible polynomials over Z, Q, R and C, and the factorisation of polynomials into irreducibles.

8. Eisenstein's criterion.

9. Characteristic. Simple extensions. Finite fields. Degree Theorem. Ruler and Compass constructions and solutions to classical geometric problems (e.g. squaring the circle).

**Reading Lists**

**Books**
**** Supplementary Text**

R B J Allenby.
*Rings, Fields and Groups*. Edward Arnold

C W Norman.
*Undergraduate Algebra*. Oxford

D A R Wallace.
*Groups, Rings and Fields*. Springer