MX4082: GALOIS THEORY (2018-2019)

• Field Theory, Field Extensions.
• Constructible Numbers.
• The Galois Group of a Field Extension.
• Cyclotomic Fields.
• Splitting Fields of Polynomials.
• Normal Extensions, Separable Extensions.
• Simple Fields Extensions.
• Counting Field Homomorphisms.
• Galois Extensions.
• The Galois Correspondence.
• Cyclic Galois Groups.
• Radical Extensions and Solvable Galois Groups.
• The Galois Group of a Polynomial. Applications.

Syllabus

• Constructible numbers.
• The impossibility of trisecting an angle and squaring the circle.
• Normal and separable field extensions. Galois extensions. Galois groups.
• The Galois correspondence.
• The Galois group of a polynomial.
• The use of Galois theory in the solution of polynomial equations.
• The impossibility of solving general polynomial equations of degree five or greater by radicals.

Course Aims

Galois theory, so named after the French mathematician Évariste Galois (1811–1832), combines field

theory and group theory to one of the highlights in algebra, yielding insight into a great diversity of mathematical problems. At the origin of this theory are questions related to the solvability of polynomial equations by radicals. We will in particular show the classical result that polynomial equations of degree at least five are not always solvable by radicals. In the process we will also obtain a description of which constructions are possible with a straight edge and a compass - proving in particular that it is impossible, in general, to trisect a given angle, to square the circle, to double the cube, or to construct a regular heptagon. The topics covered include

- Constructible numbers.

- The impossibility of trisecting an angle and squaring the circle.

- Normal and separable field extensions.

- Galois extensions.

- Galois groups.

- The Galois correspondence.

- The Galois group of a polynomial.

- The use of Galois theory in the solution of polynomial equations.

- The impossibility of solving general polynomial equations of degree five or greater by radicals.

Learning Objectives

By the end of the course the student should:

- be able to determine whether a given complex number is constructible;

- be able to calculate Galois groups of certain field extensions;

- be able to calculate Galois groups of certain polynomials;

- be able to decide whether a given polynomial equation is solvable by radicals;

- understand the Galois correspondence and more generally, theoretical aspects of the interplay between field theory and group theory applied to the structure theory of fields and solvability of polynomial equations.

Information on contact teaching time is available from the course guide.

More Information about Week Numbers

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%). Resit: If required and permitted by Regulations, there will be 1 two-hour written examination. The CAS mark will be based on the maximum of examination (100%) and examination (80%) together with in-course assessment (20%).

Informal assessment of weekly homework through discussions in tutorials.

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinators for feedback on the final examination.