This module is designed for students in their third year and aims to introduce the basic concepts and techniques of Galois theory, building on earlier work at level 2. As such, it will provide an introduction to core concepts in rings, fields, polynomials and certain aspects of group theory.
Galois theory arose out of attempts to generalize to polynomials of higher degree the well-known formula for the roots of a quadratic polynomial. This turns out to be possible for cubic and quartic polynomials but impossible for polynomials of degree five or more. This impossibility result is one of the main applications of Galois theory. Further applications to be considered are ruler-and-compass constructions; for instance, we determine all natural numbers n for which the regular n-gon can be constructed.
Much of this beautiful and fascinating theory was discovered by the French mathematician and revolutionary Évariste Galois, shortly before he was killed in a duel in 1832, aged twenty. It has
considerably influenced the development of Algebra and is nowadays a basic tool also in Number Theory and (Algebraic) Geometry. For instance, it features prominently in the famous proof of Fermat's Last Theorem by Andrew Wiles in the 1990s.
The main theorem of Galois theory gives a correspondence between the intermediate fields of a finite extension L/K of fields on the one hand and the subgroups of the automorphism group G = Aut (L / K) on the other hand. In particular, this module will introduce the concepts of rings and fields including, for example, the notions of polynomial rings, ideals, quotient rings and homomorphisms, building on material from MATH2046 Algebra and Geometry. Some group theory is also assumed, such as normal subgroups, quotient groups, and familiarity with permutation groups. These topics are all covered in MATH2003 Group Theory, which is also a pre-requisite for this module.
On successful completion of the module the students should be able to:
- show familiarity with the concepts of ring and field, and their main algebraic properties;
- correctly use the terminology and underlying concepts of Galois theory in a problem-solving context;
- reproduce the proofs of its main theorems and apply the key ideas in similar arguments;
- calculate Galois groups in simple cases and to apply the group-theoretic information to deduce results about fields and polynomials.
One of the pre-requisites for MATH3078 and MATH6156
Pre-requisites: MATH2003 AND MATH2049
Aims and Objectives
Having successfully completed this module you will be able to:
- Calculate Galois groups in simple cases and to apply the group-theoretic information to deduce results about fields and polynomials
- Correctly use the terminology and underlying concepts of Galois theory in a problem-solving context
- Reproduce the proofs of its main theorems and apply the key ideas in similar arguments;
- Show familiarity with the concepts of ring and field, and their main algebraic properties;
- Rings and Fields: Basic definitions, examples, subrings, ideals, quotient rings, isomorphism theorems, Principal Ideal Domains, Euclidean Domains and unique factorisation.
- Polynomials: The ring of polynomials; Irreducibility; Gauss's Lemma; Eisenstein's Criterion; Reduction mod-p.
- Field extensions: Degrees; Minimal polynomial and simple extensions; Algebraic and transcendental extensions; Existence and uniqueness of splitting-fields; Classification of finite fields (existence and uniqueness).
- Ruler-and-compass construction; impossibility proofs.
- Galois group: Definition, examples and basic properties; Action on roots; Separability; Order and degree of the splitting field; Towers of subfields and subgroups.
- Roots of unity: Algebraic properties; Cyclotomic extensions and cyclotomic polynomials.
- Solvability by radicals: Solvable groups; Radical extension implies solvable Galois group; Insolvability of the quintic.
- The Galois correspondence: Fixed fields; Normal extensions; Fundamental Theorem of Galois Theory.
Learning and Teaching
Teaching and learning methods
Lectures, problem sheets, tutorials, private study
|Total study time||150|
Resources & Reading list
Stewart IN. Galois Theory. Chapman & Hall.
Rotman J. Galois Theory. Springer.
Jacobson N. Basic Algebra, Volume 1, Chapter 4. Freeman.
Artin E. Galois Theory. Notre Dame.
Gaal L. Classical Galois Theory with Examples. Chelsea.
Snaith VP (1998). Algebra - Groups, Rings and Galois Theory. World Scientific Press.
Cohn PM. Algebra, Volumel 2, Chapter 3. Wiley.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External