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# Galois theory

Galois Theory – developed in the 19th century and named after the unlucky Évariste Galois, who died aged 20 following a duel – uncovers a strong relationship between the structure of groups and the structure of fields in the Fundamental Theorem of Galois Theory. This relationship has consequences, including the classification of finite fields, impossibility proofs for certain ruler-and-compass constructions, and a proof of the Fundamental Theorem of Algebra. Most famous, however, is the connection it brings between solving polynomials and group theory, culminating in the proof that there is no “quintic formula” like there is a quadratic formula.

## Qualifications

M838 is an option module in our:

## Module

Module code
M838
Credits

Credits

• Credits measure the student workload required for the successful completion of a module or qualification.
• One credit represents about 10 hours of study over the duration of the course.
• You are awarded credits after you have successfully completed a module.
• For example, if you study a 60-credit module and successfully pass it, you will be awarded 60 credits.
30
Study level
Across the UK, there are two parallel frameworks for higher education qualifications, the Framework for Higher Education Qualifications in England, Northern Ireland and Wales (FHEQ) and the Scottish Credit and Qualifications Framework (SCQF). These define a hierarchy of levels and describe the achievement expected at each level. The information provided shows how OU postgraduate modules correspond to these frameworks.
 OU Postgraduate SCQF 11 FHEQ 7
Study method
Distance learning
Module cost
See Module registration
Entry requirements

Find out more about entry requirements.

## 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 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 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.

## Teaching and assessment

• Marking your assignments (TMAs) and providing detailed feedback for you to improve.
• Guiding you to additional learning resources.
• Providing individual guidance, whether that’s for general study skills or specific module content.

The module has a dedicated and moderated forum where you can join in online discussions with your fellow students. There are also online module-wide tutorials. While these tutorials won’t be compulsory for you to complete the module, you’re strongly encouraged to take part. If you want to participate, you’ll likely need a headset with a microphone.

### Assessment

The assessment details for this module can be found in the facts box.

### Course work includes

 4 Tutor-marked assignments (TMAs) Examination

## Future availability

Galois theory (M838) starts every other year – in October.

We expect it to start for the last time in October 2034.

## Regulations

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.

## Entry requirements

You must have passed one of the following modules:

Or one of the discontinued modules M826, M828 and M832.

You should also have a good background in pure mathematics, with some experience in group theory and linear algebra – we recommend our undergraduate modules Pure mathematics (M208) and Further pure mathematics (M303).

All teaching is in English and your proficiency in the English language should be adequate for the level of study you wish to take. We strongly recommend that students have achieved an IELTS (International English Language Testing System) score of at least 7. To assess your English language skills in relation to your proposed studies you can visit the IELTS website.

### Preparatory work

Additional material is provided at the module start to help you to revise the necessary group theory and linear algebra.

## Register

Start End England fee Register
05 Oct 2024 Jun 2025 £1360.00

Registration closes 05/09/24 (places subject to availability)

This module is expected to start for the last time in October 2034.

## Future availability

Galois theory (M838) starts every other year – in October.

We expect it to start for the last time in October 2034.

### Study costs

There may be extra costs on top of the tuition fee, such as set books, a computer and internet access.

### Study events

This module may have an optional in-person study event. We’ll let you know if this event will take place and any associated costs as soon as we can.

## Ways to pay for this module

We know there’s a lot to think about when choosing to study, not least how much it’s going to cost and how you can pay.

That’s why we keep our fees as low as possible and offer a range of flexible payment and funding options, including a postgraduate loan, if you study this module as part of an eligible qualification. To find out more, see Fees and funding.

## Study materials

### What's included

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 that are contained in the set book. You’ll need to obtain your own copy of the set book.

• a week-by-week study planner to help keep you on schedule
• additional content, such as a module guide and audio and video resources.
• copies of the printed materials
• assessment details, instructions and guidance
• online tutorial access
• links to other OU websites helping you to study successfully
• an online forum for peer-to-peer interaction and interaction with the module team.

### Computing requirements

You’ll need broadband internet access and a desktop or laptop computer with an up-to-date version of Windows (10 or 11) or macOS Ventura or higher.

Any additional software will be provided or is generally freely available.

To join in spoken conversations in tutorials, we recommend a wired headset (headphones/earphones with a built-in microphone).

Our module websites comply with web standards, and any modern browser is suitable for most activities.

Our OU Study mobile app will operate on all current, supported versions of Android and iOS. It’s not available on Kindle.

It’s also possible to access some module materials on a mobile phone, tablet device or Chromebook. However, as you may be asked to install additional software or use certain applications, you’ll also require a desktop or laptop, as described above.

#### Set books

• Stewart, I.N. Galois Theory (4th edn or 5th edn) CRC Press £54.99 - ISBN 9781482245820 5th edition - ISBN9781032101583. *Note: the module was written for the 4th edition, but additional resources are provided so that the module can be studied with the 5th edition.* This item is Print on Demand, please allow 3 weeks for receipt following order.

## If you have a disability

Written transcripts of any 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.