MATH3962 Rings, Fields and Galois Theory (Advanced)

General Information

This page contains information on the Senior advanced Unit of Study MATH3962 Rings, Fields and Galois Theory (Advanced).

Taught in Semester 1.
Credit point value: 6.
Classes per week: Three lectures and one tutorial.
Lecturer(s): Adrian Nelson .

Please refer to the Senior Mathematics and Statistics Handbook for all questions relating to Senior Mathematics and Statistics. In particular, see the MATH3962 handbook entry for further information relating to MATH3962.

You may also view the Faculty Handbook entry for MATH3962 in the central units of study database.

Class Representatives

The third year pure and applied mathematics class representatives will soon meet with the coordinators to discuss all aspects of the third year units. Please email the representatives if there are any issues you would like them to raise.

View the Unit Information Sheet given out in lectures in the first week.

Homework

Note all homeworks should be submited stapled into in a manilla folder together with and completed assignment cover sheet. I strongly advise keeping a copy of any submitted work.

Homework 1 has been marked and was handed back at the end of the Thursday lecture on 29/03/2012.

First resubmissions due on or before Monday 16/04/2012. First resubmissions returned in lectures Monday 7th may.

Homework 2 has been marked and will be returned at the end of Monday's lecture 14/05/2012.

Homework 3 due at or before the beginning of the Wednesday lecture on 16/05/2012.

Unit Notes

Some Unit Notes for weeks 1 -- 11+. Posted 18/05/2012

Weekly Topics

Week 1: Motivation for the journey ahead, the Wiki article on Galois Theory .

Cardano's method for solving cubic equations. Background material on abstract algebra, composition laws, groups, monoids, homomorphisms and isomorphisms.

Week 2: You might like to look at the current Wiki article on rings.

Definition of rings, their basic properties, fields, zero divisors and integral domains. Subrings and extension rings. Modular arithmetic and residue class rings. The characteristic of a ring.

Week 3: Polynomial Rings. Evaluation maps. Homomorphism, kernels and and ideals. Ideals of the ring of integers. The field of fractions of an integral domain.

Week 4: Divisibility in integral domains, units, irreducible and reducible elements, unique factorisation domains. Principal ideal domains and unique factorisation.

Week 5: Imaginary Quadratic domains. Primes and irreducibles. Euclidean domains. The Gaussian integers and sums of two squares. Greatest common divisors. The Euclidean algorithm.

See the Wiki article on The Euclidean Algorithm .

Week 6: Equivalence relations. Composition laws and congruences. Cosets, normal subgroups and and quotient groups. Ring congruences, quotients, kernels and ideals. Isomorphism Theorems.

Week 7: Maximal ideals, simple rings and fields. Prime ideals and integral domains. Maximal ideals in principal ideal domains. Polynomial and over fields. Polynomial over the rationals, primitive polynomials and Gauss' Lemma. The Eisenstein irreducibility criteria.

Week 8: Field extension, simple field extensions, simple algebraic field extension. The degree of an extension, finite extensions, the Tower Theorem. Algebraic extensions, algebraic closure, the algebraic numbers.

Week 9: Constructing field extensions, finite field examples.Transcendental extensions, and transcendental numbers. Ruler and compass constructions, constructable numbers, duplicating the cube, squaring the circle, trisecting angles, constructing regular polygons.

Week 10: Field embeddings, splitting fields, the isomorphism extension theorem, isomorphisms of splitting fields. Galois Theory: Automorphisms, automorphism groups and fixed fields.