Galois Theory

 

Notes

 

Assuming basic knowledge of ring theory, group theory and linear algebra, these notes lay out the theory of field extensions and their Galois groups, up to and including the fundamental theorem of Galois theory. Also included are a section on ruler and compass constructions, a proof that solvable polynomials have solvable Galois groups, and the classification of finite fields. Tom Leinster, Galois Theory, arXiv:2408.07499, 2024. You can also read the slides from the introductory lecture.

 

Videos

 

I first taught the course in 2021, during full Covid lockdown. Anyone who taught during that strange era — and especially anyone who created a course then — has the experience seared onto their memory. The workload was colossal. We suddenly had to abandon the face-to-face teaching methods we'd spent years fine-tuning and come up with entirely new ways of doing things — figuring out everything from hardware installation to video sharing platforms, from online assessment to the difficult question of how to create a sense of belonging and human connection in a virtual classroom of isolated individuals. And all this at the same time that each of us was coping with our own personal challenges in the suddenly locked down world. I'm not particularly proud of these videos. When I watch them now, I can hear the exhaustion in my voice. And for the sake of getting at least a little sleep, I stuck to the rule that the first take would be the only take. But I think it's worth sharing them, so here they are. Warning!  As the videos were made in 2021, they refer to that year's notes, which were similar but not identical to the final version linked above. In particular, some theorem numbers changed. Hopefully it won't be too hard to figure out what's going on when I use theorem numbers in the videos. Where the titles of the videos contain theorem numbers, I have updated them to the numbering used in the final version. Chapter 1: Overview of Galois theory Introduction to Week 1 Example 1.1.5 Example 1.2.2 Galois groups, intuitively Chapter 2: Group actions, rings and fields Introduction to Week 2 The meaning of "n.1", and Exercise 2.3.13 Quotient rings. Correction: where I write "r, r' ∈ I" near the right of the page, it should be "r, r' ∈ R" Building blocks A strictly ascending chain of ideals Chapter 3: Polynomials Introduction to Week 3 Why study polynomials? The video mentions this MathOverflow question The universal property of R[t] Exercise 3.2.4: a non-principal ideal Testing for irreducibility Chapter 4: Field extensions Introduction to Week 4 Two traps How to understand simple algebraic extensions Chapter 5: Degree Introduction to Week 5 Ruler and compass constructions Chapter 6: Splitting fields Introduction to Week 6 (and the rest of the course) Extension problems Explanation of Lemma 6.1.3. There's a problem with the screenshare in this video. Download this PDF to see what I'm writing Counting isomorphisms: the proof of Proposition 6.2.11 Calculating Galois groups with your bare hands, part 1 Calculating Galois groups with your bare hands, part 2. The screenshare freezes two minutes from the end, so you can't see the last four lines I write, but this PDF file shows them The action of the Galois group Chapter 7: Preparation for the fundamental theorem Introduction to Week 7 What does it mean to be normal? Splitting field extensions are normal: Theorem 7.1.5. At 2:20, I say "splitting fields are easiest when you're talking about a polynomial over Q, because then the splitting field is simply Q with the roots of your polynomial adjoined." That's true for polynomials over any field, not just Q. What I meant to say is that when you're over Q, you can understand the roots to be complex numbers and the splitting field to be a subfield of C. So you can see it more concretely over Q The size of fixed fields Chapter 8: The fundamental theorem of Galois theory Introduction to Week 8 Finding fixed fields Normal subgroups and normal extensions Chapter 9: Solvability by radicals Introduction to Week 9 The definition of radical number Solvable polynomials have solvable groups: a map, and PDF of the map itself Chapter 10: Finite fields Introduction to Week 10 The multiplicative group of a finite field is cyclic Ordered sets

 

Problems

 

The course included fortnightly workshops (otherwise known as tutorials or exercise classes) and written assignments. Because of strikes and timetable changes, they covered slightly different topics in the three years that the course ran. I'm providing the materials from all three years. Where the notes refer to workshops and assignments, it's the 2023 versions. I will not be sharing solutions to the workshop or assignment questions. 2023 Workshop 1: Revision; overview of Galois theory Assignment 1 Workshop 2: Overview of Galois theory; group actions, rings and fields; polynomials Assignment 2 Workshop 3: Polynomials; field extensions; degree Assignment 3 Workshop 4: From splitting fields to the fundamental theorem Assignment 4 2022 Workshop 1: Overview of Galois theory; group actions, rings and fields Assignment 1 Workshop 2: Polynomials; field extensions Assignment 2 Workshop 3: Degree; splitting fields Assignment 3 Workshop 4: Preparation for the fundamental theorem Assignment 4 2021 Workshop 1: Overview of Galois theory; rings and fields Assignment 1 Workshop 2: Polynomials; field extensions Assignment 2 Workshop 3: Degree; splitting fields Assignment 3 Workshop 4: The fundamental theorem of Galois theory Assignment 4 Workshop 5: Solvability by radicals; finite fields Assignment 5

 

Quizzes

 

The course first ran during Covid lockdown, so all the students were learning in total isolation, deprived of the usual regular contact with their peers. I was therefore concerned that it would be hard for them to gauge whether they were making good progress. To address this, I created an online multiple choice quiz which the students could take any time they felt like it, and as many times as they liked — just for fun, not for credit. Each time they took it, the software would randomly choose ten questions on the material so far, drawn from a bank of nearly 500 questions. The questions were hosted on the course STACK site and written in the very useful Latex moodle package, which generates Moodle XML. This package can also generate PDF versions, as below. Here are the PDF files of the quizzes. Again, I will not be sharing the solutions, which means I also won't be sharing the Latex source files. All 477 questions Quiz 0: Revision Quiz 1: Overview of Galois theory Quiz 2: Group actions, rings and fields Quiz 3: Polynomials Quiz 4: Field extensions Quiz 5: Degree Quiz 6: Splitting fields Quiz 7: Preparation for the fundamental theorem Quiz 8: The fundamental theorem of Galois theory Quiz 9: Solvability by radicals Quiz 10: Finite fields

 

This page was last changed on 20 August 2024. Home