Galois theory for PROMYS, 2024

This is the website for the second year course "Galois Theory" at PROMYS 2024. The class will be highly interactive, taught through collaborative problem solving. This webpage records the problems we worked on in class, as well as the homework assignments.

Worksheets

(July 1) Worksheet 1 — The quadratic formula and Worksheet 2 — The cubic formula

(July 1) We solved all problems and had a number of interesting side discussions. I worry that the speed may have been intimidating to some; let me reassure you that we will circle back to everything, in some cases on the next day, and in other cases in days to come.

(July 3 and July 4) Worksheet 3 — Groups of permutations, and groups in general

(July 3) We understood the concepts of orbit and stabilizer, and stated the orbit stabilizer theorem. We'll prove it next time.
(July 4) We proved the orbit-stabilizer theorem. I'll record the key idea of the proof here. Let a group G act on a set X and let x be an element of X. For each y in the orbit of x, let G(x → y) be the set of group elements g such that gx=y. We showed that each of the sets G(x → y) had the same size, and so the size of G is the number of these sets (namely, the number of orbits) times their common size (namely the size of the stabilizer).

(July 4) Worksheet 4 — Subgroups, cosets, orders of elements

(July 4) We introduced the notion of cosets, and used them to prove the more general Lagrange's theorem: For G a group and H a subgroup, the size of G is the product of the size of H and the number of cosets of H in G. In particular, |H| divides |G|.
I have mixed feelings about how this class went. The discussion went very fast because some students were roaring ahead, and I fear that others were left behind. If I were teaching a normal group theory course, I'd definitely spend another day here to help everyone catch up. But I have limited time, and the main thing that I will want is group actions and the orbit-stabilizer theorem, not the more abstract cosets and Lagrange's theorem. I'm going to move on, but I've put some problems on problem set 3 to address this, and I would be glad to discuss them with you.

(July 8, 10 and 11) Worksheet 5 — Characters of groups and Worksheet 6 — Characters coming from polynomials.

(July 8) We finished problems 5.1 and 5.2, and had an excellent discussion of Problem 6.1. We'll return to 6.1 on Wednesday, as well as the later problems on sheet 6. We may or may not return to problem 5.3. If you'd like to think about that problem, here is a hint: First figure out what χ does for reflections.
(July 10) We gave a full proof of 6.1.(2), by directly analyzing how ε(w) and ε(tw) relate for t a transposition. What I really like about this proof is that it is very direct; slogging ahead in a straightforward way. Expect a slicker proof when we return to Worksheet 6 tomorrow.
(July 11) We gave a full proof of all the results on Worksheet 6. I really liked the suggestion to break 6.4 into two parts: (1) If g stabilizes F^k, then gF is a scalar multiple of F and (2) if F is nonzero H is a group such that hF = χ(h) F for all h in H, then χ is a character of H.
I didn't get a chance to point this out, but note that this is a different, and in my opinion much slicker, proof that ε is a character.

(July 15 and 17) Worksheet 7 — Characters of the symmetric and alternating groups

(July 15) We classified the characters of S_n. I also hope we made some new friends! Tomorrow, we'll classify the characters of A_n.
(July 17) We showed that, for n ≥ 5, the alternating group A_n has no nontrivial characters. This positions us to prove a big theorem tomorrow.

(July 18) Worksheet 8 — Unsolvability of the quintic, first version

(July 18) Great class! We proved the theorem! For some of you, this is the end; for others, see you in the second half of the course!

(July 22 and 24) Worksheet 9 — Lemmas about polynomials

(July 22) We made reasonable progress, getting through problem 9.3. We'll finish this up on Wednesday.
(July 24) We finished all the problems, and also introduced the terminology of "algebraic numbers" and the "fundamental lemma of linear algebra". We'll carry these ideas into the next worksheet tomorrow.

(July 25, 29 and 31) Worksheet 10 — Algebraic elements and Worksheet 11 — Minimal polynomials.
- (July 25) We proved all the results on Worksheet 10, and started talking about Worksheet 11.
- (July 29) We got through the first two problems very solidly and rushed our way through the second two. We'll come back to those two a little on Wednesday but mostly move on.
- (July 31) We had a good discussion of how to use Problem 11.4.(2) to construct isomorphisms between different fields, and even to construct automorphisms of fields!
(July 31, August 1) Worksheet 12 — Splitting fields.

(July 31) We proved the existence of splitting field. Uniqueness needs to wait for tomorrow.
(August 1) We proved uniqueness, and also proved the very useful Problem 12.4, which will continue to be useful.

(August 1 and August 5) Worksheet 13 — Automorphisms of splitting fields.

(August 1) Great high energy class! I wish I had more time to let the discussion roll. We did 13.1, 13.2 and 13.4.(1) on the board, and I know many of you were much further ahead than that. We'll pick up on Monday.
(August 5) Students seemed tired today. We finished all the problems, including 13.5 and 13.7 which are big ones. I hope that people followed the arguments; they are short but surprisingly powerful.

(August 7) Worksheet 14 — The commutator subgroup and, with luck, start on Worksheet 15 — Unsolvability of the quintic, second version.

(August 7) We proved all the problems about commutators, and started discussing our strategy for the big theorem.

(August 8) Worksheet 15 — Unsolvability of the quintic, second version.

(August 8) We proved everything! Then we talked about how to use Mathematica to compute some (small) Galois groups. Here is the Mathematica notebook we used. If I get a chance, I'll go through and add comments but, realistically, this may not happen.

Problem Sets

Problem set 1, due July 4
Problem set 2, due July 8
Problem set 3, due July 11
Problem set 4, due July 15
Problem set 5, due July 18
No problem set due July 22; work on your midterm write up and catch up on sleep.
Problem set 6, due July 25
Problem set 7, due July 29. Note: There are a lot of interesting problems here, but none of them are essential; you can keep going without solving these.
Problem set 8, due August 1. Note: The problem on doubling the cube is ``side quest", we won't come back to it.
Problem set 9, due August 5. Note: The problems on commutators will be relevant very soon; all the others are side quests.
Problem set 10, due August 8. Note: Problem 1 will be key on the last day. The others are all excellent side quests.
Open Door Set!