What you will study
In order to prove the Fundamental Theorem of Galois Theory and unlock its many applications, you’ll first study the fundamentals of abstract algebra. These include the study of groups, rings and fields, homomorphisms (that allow you to move between two structures) and automorphisms (that allow you to rearrange the elements within a structure in a controlled way). Some prior group theory and linear algebra is assumed in the early units, but supplementary material is available if you need to refresh your knowledge.
This module is based on readings from the set book, Galois Theory (4th edition) written by Ian Stewart, but the study materials contain additional reading both before you start the set book, and at the end.
After studying the fundamentals of algebra, you’ll learn about field extensions, which is a way of thinking about two fields, one of which is contained inside the other. Each extension comes with a set of automorphisms of the bigger field that leave the smaller one fixed. They form a group, called the Galois group, and here you will get your first glimpse of what is to come: the Fundamental Theorem of Galois Theory is about the relationship between the structure of a field extension and the structure of its Galois group.
Throughout, you’ll see how this study of groups and fields is intimately connected to the solution of polynomials. For example, a polynomial that has no solutions in the field of real numbers can always be split up into linear terms if you use the field of complex numbers. Thus, there is a relationship between field extensions and solving polynomials. You’ll learn about two properties – normality and separability – that a field extension may or may not possess, that are defined in terms of how polynomials split up in the bigger field. These two properties are the final ingredients required for you to prove the Fundamental Theorem of Galois Theory.
The last part of the module covers the applications of the Fundamental Theorem. By far the most famous is the proof that while there exists a formula to solve any quadratic equation (and also formulas to solve cubic and quartic polynomials, though these are less well known and not of much use in practice), there is no such way to solve polynomials of degree 5 or greater. You’ll see how the Fundamental Theorem can be used to translate this problem into one about group theory, which can be solved with relative ease.
Other applications include proofs for the existence or non-existence of ruler-and-compass constructions for regular polygons, a complete classification of finite fields, and at the very end, an algebraic proof of the Fundamental Theorem of Algebra.
You will learn
Successful study of this module should enhance your skills in understanding complex mathematical texts, working with abstract concepts, thinking logically and constructing logical arguments, communicating mathematical ideas clearly and succinctly, and explaining mathematical ideas to others.