[The University of Leicester]

Department of Mathematics



Next: MA4501 Mathematics Project Up: ModuleGuide03-04 Previous: MA4151 Lie Algebras

MA4161 Galois Theory

MA4161 Galois Theory

Credits: 20 Convenor: Prof. S. König Semester: 2
Prerequisites: essential: MC241, MC255 desirable: MC382
Assessment: Coursework: 10% Three hour examination in June: 90%
Lectures: 36 Problem Classes: 10
Tutorials: none Private Study: 104
Labs: none Seminars: none
Project: none Other: none
Surgeries: none Total: 150

Subject Knowledge

Aims

To help the student to develop an appreciation of the rigorous development of this beautiful subject, and an understanding of the fundamental results of the subject.

Learning Outcomes

To enable the student to :

Understand and define the Galois group of a field extension, and the Galois group of a polynomial.

Understand and prove the Galois correspondence, including the relationship between the normal subject structure of the Galois group and normality of intermediate extensions.

Understand the definition of a solvable group and be able to determine whether or not a group of reasonable size is solvable.

Appreciate the significance of the Galois group of a polynomial as a group of permutations of the roots.

Prove that the alternating group of degree at least 5 is simple.

Understand the definition of a radical extension and prove that such an extension has a solvable Galois group.

Understand Lagrange resolvents and their use in proving that a finite separable normal extension with a solvable Galois group is a radical extension.

Understand that the symmetric group is the Galois group of the general polynomial.

Be able to construct polynomials whose Galois group is not solvable.

Subject Skills

Aims

This course should assist the student to develop skills of careful and rigorous Mathematcial writing and expression. It could also serve as preparation for s student wishing to embark on research in Pure Mathematics, as it should convey an appreciation of a common strand of development in modern Mathematics.

Explanation of Pre-requisites

It will be assumed that the student has some familiarity with the basic properties of groups, and knows the beginnings of the theory of field extensions. It would be desirable, but is not essential, for the student to have seen the definition of a solvable group.

Course Description

Galois theory is one of the first examples of methods from one branch of Mathematics being applied to solve problems in an apparently completely different area. Which regular $n$-sided polygon can be constructed by ruler and compass? For a specific $n$, such a question can be given a positive answer by finding an explicit construction. But how does one prove that there is no such construction for certain $n$? And what is the difference between $n$'s for which there is a construction and $n$'s for which there isn't?

Surprisingly, this question is closely related to the problem of solving polynomial equations by radicals, that is, finding a formula which only involves the coefficients of the polynomial and arithmetic operations including taking roots. Everybody knows the answer for polynomials of degree two. But what happens in general? Again, for a specific $n$ one may give a positive answer by finding an explicit formula. But how does one prove that there is no such formula for certain $n$?

This course will provide complete anwers to both problems by transforming them into questions about algebraic structures such as groups and fields. The main theorem of Galois theory is one of the most beautiful theorems in all of mathematics, and extensions and applications of Galois theory are the subject of major research activities in algebra, geometry and analysis.

Syllabus

Automorphisms of field extensions. The Galois group of a finite extension. Normal extensions. Separability. The Galois correspondence. The Galois group of a polynomial, viewed as a group of permutations of the roots. Transitivity of the Galois group of an irreducible polynomial on the roots. Composition series for groups and characterization of solvable groups in terms of these. Alternating groups and proof of their simplicity in degree greater than 4. Radical extensions and the solvability of their Galois groups. Lagrange resolvents and the proof that a radical normal extension has a solvable Galois group. The symmetric group as the Galois group of the ``general''polynomial. Explicit determination of the general solution of the general cubic and quartic. Examples of polynomials which are not solvable by radicals.

Reading list

Background:

P.M.Cohn, Algebra II,

S.Lang, Algebra,

J. Rotman, Galois Theory,

I. Stewart, Galois Theory, Chapman and Hall.

Module Evaluation

Module questionnaires, module review, year review.

Next: MA4501 Mathematics Project Up: ModuleGuide03-04 Previous: MA4151 Lie Algebras

[University Home] [MCS Home] [University Index A-Z] [University Search] [University Help]

Author: C. D. Coman, tel: +44 (0)116 252 3902
Last updated: 2004-02-21
MCS Web Maintainer
This document has been approved by the Head of Department.
© University of Leicester.