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 are assumed in the early units, but supplementary material is available to refresh your knowledge.
This module is based on readings from the set book, Galois Theory (4th or 5th editions), 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 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 its Galois group.
Throughout, you’ll see how this study of groups and fields intimately connects to the solution of polynomials. For example, a polynomial with no solutions in 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, which are defined in 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.
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.
You’ll get help and support from an assigned tutor throughout your module.
They’ll help by:
Online tutorials run throughout the module. While they’re not compulsory, we strongly encourage you to participate. Where possible, we’ll make recordings available.
Course work includes:
You’ll be provided with course notes covering the content of the module, including explanations, examples, and exercises to aid your understanding of the concepts and associated skills and techniques contained in the set book. You’ll need to obtain your own copy of the set book.
You’ll also have access to a module website, which includes:
Set books:
You can study this module on its own or use the credits you gain towards an Open University qualification.
M838 is an option module in our:
Galois theory (M838) starts every other year – in October.
It will next start in October 2026.
We expect it to start for the last time in October 2034.
As a student of The Open University, you should be aware of the content of the academic regulations, which are available on our Student Policies and Regulations website.
You must have passed (or be studying in parallel) one of the following modules:
You should have a solid background in pure mathematics, with some experience in group theory and linear algebra. Pure mathematics (M208) and Further pure mathematics (M303) should provide adequate preparation.
We provide additional material at the module start to help you revise the necessary group theory and linear algebra.
Written transcripts of audio components and Adobe Portable Document Format (PDF) versions of printed material are available. Some Adobe PDF components may not be available or fully accessible using a screen reader (and where applicable, musical notation and mathematical, scientific, and foreign language materials may be particularly difficult to read in this way). Other alternative formats of the module materials may be available in the future.
To find out more about what kind of support and adjustments might be available, contact us or visit our disability support pages.
| Start | End | Register by | England fee |
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| 03 Oct 2026 | 30 Jun 2027 | 10 Sep 2026 | Not yet available* |
| *This start date is open for pre-booking, which means you can reserve your place ahead of the fees being confirmed. We’ll publish updated 2026/27 fees and funding information in late March 2026. |
There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.
If your income is not more than £25,000 or you receive a qualifying benefit, you might be eligible for help with some of these costs after your module has started.
There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.
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If you study this module as part of an eligible qualification, you can apply for a postgraduate loan to help with your tuition fees. To find out more, see Postgraduate loans in Scotland.
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