This module focuses on fields, their extensions, and their deep connection to the theory of groups. It thus provides an introduction to Galois theory, one of the most fascinating branches of algebra, developed by the young French 19^th-century mathematician Évariste Galois shortly before his tragic death in a duel at the age of twenty. Galois theory provided satisfying answers to some of the oldest mathematics questions in the world: the problems of duplicating a cube or trisecting an angle date back to antiquity, while the problem of solvability of quintic equations dates back to the Renaissance.   

In this Year Three module, students will be introduced to the basic notions of the theory of  field extensions, the key results of Galois theory, and their well-known applications to ruler-and-compass constructions and to the solvability of polynomial equations.

Prerequisite Modules: MA2013 - Algebra I: Groups,
Recommended Modules: M3013 – Algebra II: Rings

1. State the definition of a field and a field extension.
2. Understand the basic notions of the theory of field extensions: the degree of an extension, algebraic and transcendental elements, minimal polynomials, splitting extensions, normal extensions, separable extensions.
3. Understand the main notions and results of Galois theory: the fixed field of a field automorphism, the Galois group of a field extension, and the relation between the two.
4. Apply the results of Galois theory to ruler-and-compass constructions and to the solvability of polynomial equations by radicals.

Compute the Galois groups of simple polynomials.

Modules will be delivered through blended learning. You will be guided through learning activities appropriate to your module, which may include:

• Weekly face to face classes (e.g. labs, lectures, exercise classes)
• Electronic resources that you work through at your own pace (e.g. videos, exercise sheets, lecture notes, e-books, quizzes)

Students are also expected to undertake self-guided study throughout the duration of the module.

Skills:
Appreciation of the general prevalence and significance of fields in mathematics. Ability to understand and apply the fundamental concepts of Galois theory.

Transferable Skills:
Recognition of the power of mathematical abstraction and the importance of mathematical rigour. Critical thinking and problem-solving skills. Ability to present work in a scholarly manner.

1. Fields and field extensions. Degree of an extension. Algebraic and transcendental elements. 
2. Splitting field of a polynomial. Splitting field extensions. Algebraic closure. 
3. Normal extensions. Separable extensions.
4. Galois theory: the fixed field of a set of field automorphisms and the Galois group of a field extension. The Galois group of a polynomial. The fundamental theorem of Galois theory.
5. Applications: ruler-and-compass constructions, solvability by radicals, computation of Galois groups