MATH40005 Galois Theory

This is the former 451/MA4005/MT4161)

SEMESTER: First
CONTACT: Dr Hovhannes Khudaverdyan (M/P2) CREDIT RATING: 15
Aims: To introduce students to a sophisticated mathematical subject where elements of different branches of mathematics are brought together for the purpose of solving an important classical problem.
Intended Learning Outcomes: On successful completion of the course students will:
  • Have deepened their knowledge about fields
  • Have acquired sound understanding of the Galois correspondence between intermediate fields and subgroups of the Galois group
  • Be able to compute the Galois correspondence in a number of simple examples
  • Appreciate the insolubility of polynomial equations by radicals.
Pre-requisites: 212,252,312 (ex-UMIST)
MT2262, UM3121 (ex-VUM)
Dependent Courses: None
Course Description: Galois theory is one of the most spectacular mathematical theories. It gives a beautiful connection of the theory of polynomial equations and group theory. In fact, many fundamental notions of group theory originated in the work of Galois. For example, why some groups are called "soluble" ? Because they correspond to the equations which can be solved ! (Meaning by a solution some formula based on the coefficients and involving algebraic operations and extracting roots of various degrees.) Galois theory explains why we can solve quadratic, cubic and quartic equations, but no similar formulae exist for equations of degree greater than 4. It also gives complete answer to ancient questions such as divising of a circle into n equal arcs using ruler and compasses. In modern exposition, Galois theory deals with "field extensions", and the central topic is the "Galois correspondence" between extensions and groups. Galois theory is a role model for mathematical theories dealing with "solubility" of a wide range of problems.
Teaching Mode: 2 Lectures per week
1 Tutorial per week
Private Study: 5 hours per week
Recommended Texts: I Stewart, Galois Theory, 2nd edition, Chapman and Hall
J B Fraleigh, A First Course in Abstract Algebra, 5th edition, 1967, Addison-Wesley
Assessment Methods: Examination: 80%

A 2 hour examination at the end of the Second Semester.

Coursework: 20%

One homework assignment set in week 5, deadline in week 8

No of lectures Syllabus
4 Background and motivation.Viete Theorem. Symmetric functions of roots of polynomials. Group of permutations of roots. Ring of polynomials. Irreducibility tests.
6 Field extensions.Finite extensions. Splitting fields and normal extensions. Theorem about existence of primitive element.
6 Galois groups. Resolvent polynomials. Fundamental Theorem of Galois Theory.
4 Quadratic irrationalities. Galois group and constructions by ruler and compasses. Construction of a regular n-sided polygon.
4 Solution of polynomial equation in radicals and solubility of the Galois group . Cubics and quintics.
   Last revised August, 2006