This semester I am teaching the Galois Theory course. The exercise classes will be given by Milan Lopuhaä, who has provided lots of useful information on the course here. Useful information can also be found on the prospectus page for this course.
Week 1 (02/09):
Recap on field theory, including definitions of field extensions, algebraic and transcendental elements, and homomorphisms of field extensions.
A pdf of the lecture notes so far is coming soon!
Week 2 (09/09):
We recalled two important examples of field extensions: fields obtained by adjoining a root of an irreducible polynomial, and splitting fields of polynomials. We discussed some properties of Hom_K (L,M) when L|K is one of these extensions. In Keune's book, this roughly corresponds to the end of chapter 1 and the first half of chapter 4.
** UPDATE ** Correction to sheet 2 (corrected version is attached)
Week 3 (16/09):
We defined the notion of a Galois extension and the notion of a normal extension (and say a bit more about that). We discussed some examples, in particular the case of finite fields. In Keune's book, we finished chapter 4 and did most of chapter 3.
If you have any questions about the lectures or the exercises, feel free to send me an email! I'd especially recommend that if after you've gone to the exercise class, handed in your answers, and got your work handed back you're still confused about a question that you can always email a question about that question and what you might have been missing in your answer.
Week 4 (23/09):
We studied normal extensions, separable polynomials and separable extensions. We also examined some Galois extensions in detail and saw how to calculate Galois groups explicitly.
Sorry for the confusion about the schedule for the homework this week.
Week 5 (30/09):
We finished our study of separable extensions and proved the Primitive Element Theorem.
Week 6 (07/10):
We proved some useful properties of separability and stated the Fundamental Theorem of Galois Theory. We then studied cyclotomic extension of Q.
** UPDATE ** Correction to sheet 6 (corrected version is attached)
Week 7 (14/10):
We gave an explicit description of the Galois groups of cyclotomic fields.
Week 8 (21/10):
We proved the fundamental theorem of Galois theory, and introduced the historical motivation for proving it.
In celebration of Evariste Galois's 204th birthday, this week the exercise sheet was slightly shorter.
There was no lecture on 11/11.
Week 9 (18/11):
We studied symmetric polynomials and used them to construct interesting Galois extensions.
Week 10 (25/11):
We introduced radical extensions and solvable groups. We saw that radical Galois extensions have solvable Galois groups.
** UPDATE ** Correction to sheet 10 (corrected version is attached)
Week 11 (02/12):
We proved that the general quintic equation is not solvable. We started thinking about how to determine Galois groups of extensions
of Q, and stated a very useful theorem about relating the Galois group of the splitting field of a polynomial to properties of its reduction
mod p.
Week 12 (09/12):
We saw how to determine Galois groups of extensions of Q of small degree. We then discussed Kummer theory and the normal basis theorem.
Week 13 (16/12):
We finished the examinable material, and then discussed Artin's approach to Galois theory. We introduced infinite Galois theory (which is only examinable for Masters students).
In view of the proximity of Christmas, this week the exercise sheet was slightly shorter.
** UPDATE ** 20/12/2015 Changed question 1, added a new question.
Week 14 (06/01):
We did some revision and then talked about profinite groups.
Week 15 (13/01):
We did some revision and then talked about infinite Galois theory, the infinite Galois correspondence and profinite completions.