Galois Theory III (MATH3041)
General Information
Welcome to Galois Theory III and one of the most beautiful and elegant subjects in mathematics!
In this module, there will be two Lectures per week plus a Problems Class every fortnight (check your timetables for details). A rough outline of what we’ll be doing in Michaelmas term is as follows.
In the first three lectures we’ll see some ideas motivating the subject. Did you ever want to solve the equation \(x^4+2x^2-16x+17=0\)? This is your chance to find out how.
The next three lectures will include the most important facts about fields and polynomials, with special attention paid to detecting irreducibility.
We’ll then spend four lectures on constructions of field extensions followed by four lectures introducing the concept of a Galois extension.
Then there will be a few lectures introducing Galois groups leading up to the Fundamental Theorem of Galois Theory. The term will conclude with a lengthy series of computations with Galois groups, which should be considered as the primary achievement of the work so far.
In Epiphany term, things will continue with the main focus being on applications in Number Theory and the Theory of Polynomial Equations. In particular, we will study
The Galois Theory of finite extensions of Finite Fields.
Cyclotomic extensions, which arise by adjoining roots of unity to a field.
Cyclic extensions, which are Galois extensions with cyclic Galois group. We will give a full description of these (Kummer Theory), when the base field contains suitable roots of unity.
Galois groups of polynomials. In particular, we will give a full description in the cubic and quartic cases, as well as showing Galois’ Theorem which gives a criterion for when polynomial equations can be solved by radicals.
Background Material from Algebra II
You all took Algebra II previously and perhaps you found it easy, but perhaps not. To get anywhere in Galois Theory requires a sound understanding of basic algebra concepts (polynomials, rings, fields, and groups) and, to ease you in, there’s a Chapter zero in the lecture notes recapping the Background Material we’ll need. There is an extra Problem sheet zero accompanying this to check how much you’ve remembered from Algebra II. There won’t be explicit lectures covering this background chapter but I’ll try to remind you of the most important concepts as and when we need them. To keep on top of things, you should definitely spend a bit of time looking through
the “Rings and polynomials” section and problems before lecture 4,
the “Group Theory” section and problems before lecture 14
to be fully ready for what follows.
Problem Sheets and Assignments
In each term, there will be 4 Problems Sheets to test your understanding of the material as we go along. As we’ll be taking a particularly computational approach to the topic over the year, it will be very important to attempt these to get a grip on things…
For some extra help, we’ll walk through some problems together in the fortnightly Problems Classes. Other problems will be set in 4 written Assignments over each term, to be submitted via Gradescope (following the link from the module Ultra page as usual). These will be marked (promptly I hope!) and will include some feedback for your perusal and edification.
Recommended textbooks
As Galois Theory is a classical subject, there are many books available to supplement the lectures, notes and problems. There are several listed in the Library Reading List and of particular note are
“Galois Theory through exercises” by Juliusz Brzezinski. This covers pretty much the same material as us and we’ll aim for a similar balance between theoretical and computational aspects. We have access to an online version so you don’t even have to go to the library…
“Galois Theory” by Ian Stewart. This is a gentle introduction with lots of historical detail - recommended bedtime reading!
As well as textbooks, there are plenty of online resources available that you’ll find with a quick internet search. For instance, Wikipedia generally has articles giving a decent overview of concepts and examples (though not a substitute for detailed lecture notes and/or textbooks).