MAU34101 - Galois theory

Course information

Lecturer: Nicolas Mascot

Assessment: 15% homework assignments, 85% final exam.

Because of the current situation, for now, all the lectures will be prerecorded and available online on this page; see below.

Lecture notes

Lecture notes for this module are available here.

Alternatively, you may want to take a look at Ian Stewart's book on Galois theory.

Recorded lectures

This section will be regularly updated until the end of the term.

The course contents are available as prerecorded videos and as PDF notes. You can save these files by right-clicking on the links and clicking on "Save link as".

Chapter 0: Introduction - What is Galois theory about?
Video (17:44) - Slides

Chapter 1: More on field extensions
Notes (Complete)
Videos:
- Reminders on algebraic extensions (08:55)
- Stem fields, splitting fields (51:29)
- Galois conjugacy (12:44)
- Finite fields (1/4): characteristic (10:49)
- Finite fields (2/4): Frobenius (06:57)
- Finite fields (3/4): classification (28:39)
- Finite fields (4/4): explicit construction (11:31)
- Polynomials and their roots (17:55)
- Resultants (31:00)
- Discriminants (14:28)

Chapter 2: The Galois correspondence
Notes (Complete)
Videos:
- More on automorphisms (29:52)
- Separable extensions (29:50)
- Separability vs. embeddings (39:27)
- Reminders on group actions (09:54)
- Normal extensions (29:28)
- Galois extensions (20:34)
- Galois groups (47:05)
- The Galois correspondence: Statement and proof (33:53)
- The Galois correspondence: Practice by examples (53:44)
- Application to cyclotomy (50:06)
- Other applications (non-examinable) (26:06)

Recorded in-person lectures

We are back to in-person lectures since the reading week. Here are the recordings of these lectures.

Monday November 01
Tuesday November 02 (I forgot to start recording at the beginning of the lecture, my apologies! But I still recorded most of it. What's missing is essentially the formula disc(x³+bx+c)=-4b³-27c³.)
Wednesday November 03
Monday November 08
Tuesday November 09
Wednesday November 10
Monday November 15
Tuesday November 16
Wednesday November 17 (Unfortunately, this recording has no sound.)

Here are slides summarising the contents of the in-person lectures; these slides are far less detailed that the ones for online lectures, and were written purely to help with revisions for the exam.

Chapter 3: The Galois group of a polynomial
Chapter 4: Solvability by radicals

Homework

You must submit your homework assignments before the deadline, by emailing it to me.

The solutions will be made available immediately after the deadline, so late submissions will not be accepted.

Exercise sheet	Due date	Solutions
Sheet 1	Wednesday October 6, 4PM	Solutions 1
Sheet 2	Friday October 22nd, 4PM	Solutions 2
Sheet 3	N/A	Solutions 3
Sheet 4	Monday November 22nd, 4PM	Solutions 4

Exam

Here is a mock exam paper for you to practice on, and here are the solutions.

I still do not know whether the final exam will take place in person or online...

Back to Nicolas Mascot's home page