Galois theory for PROMYS, 2021

At PROMYS 2021, I taught a course for advanced students on Galois theory. This course was taught through in class problem solving and, due to COVID-19, was conducted online. Here is a record of the worksheets and problem sets we used.

Worksheets

• (July 5) Worksheet 1 -- The quadratic formula
  • Status: We have solved all problems.
• (July 6) Worksheet 2 -- The cubic formula
  • Status (July 5): We started discussing Problem 2.1.
  • Status (July 6): We finished all the problems, and now understand the cubic formula!
• (July 7) Worksheet 3 -- Groups of permutations
  
  • Status (July 7) Problems 3.1-3.3 went well, and many groups also solved 3.4. We'll see 3.4 more generally next time, so we'll return to this issue then. I'll leave 3.5 for you to do on your own time (but I'm glad to help if you need help); it isn't hard. I have fixed the typo in the definition of Stab(x).
• (July 8) Worksheet 4 -- Groups, orbits and stabilizers 
  • Status (July 8) This gave people an opportunity to really make sure they understood the notions of orbit and stabilizer in examples; by the end of class, I think most people did. On Monday, we'll finish off the orbit-stabilizer theorem before moving on.
• (July 12) Worksheet 5 -- Subgroups, orbits, orders of elements Also expect a fair bit of time spent on the orbit-stabilizer theorem and homework.
  • Status (July 12) We finished proving the orbit-stabilizer theorem and used it as an opportunity to discuss Problem 5 from Problem Set 1. We then moved to talking about cosets and applying them to prove Fermat's little theorem. This was rushed, but some good conversation happened, and we will return to it tomorrow.
  • Status (July 13) We proved that the order of a subgroup always divides the order of a group, and applied this to prove Fermat's Little Theorem. The following question was posed as one to ponder: Let G be a group and H is a subgroup. Is there always a set X, with an action of G, and an element x of X, such that H = Stab(x)?
• (July 13) Worksheet 6 -- Characters of groups
  • Status (July 13): A great class! Problems 6.1 and 6.2 went very well and good starts were made on 6.3. Problem 6.3 will reappear (with hints) on Problem Set 4.
• (July 14) Worksheet 7 -- Characters coming from polynomials
  • Status (July 14): People did a lot of examples, and individual groups made a lot of progress, but we had to finish before everything came together. We'll return to this.
  • Status (July 15): We went through all the problems and discussed them.
• (July 15, July 19): Worksheet 8 -- Characters of the symmetric and alternating groups
  • Status (July 15): We spent most of the day catching up, but also got a chance to learn about conjugacy classes and start Problem 8.1.
  • Status (July 19): An extremely productive day! We all classified the characters of the symmetric group, and many of us got through the alternating group as well. We'll finish off the alternating group tomorrow.
• (July 20) We reach our first triumph: Worksheet 9 -- Unsolvability of the quintic, first version
  
  • Status (July 20): Victory is ours! We proved everything! The version here on the website fixes the duplicate question from the class version, and also adds an alternate definition of the character chi that some people found helpful.
• (July 21, July 22) Worksheet 10 -- Lemmas about polynomials .
  • Status (July 21) We first picked up two loose threads from our previous discussion: Explaining why we restrict ourselves to rational funcitons of the roots instead of just all functions, and proving that S_n cannot act with an orbit of size m for 2<m<n (when n is at least 5). I mentioned in class that there were many proofs of the simplicity of A_n, but they were all a bit fiddly; see Keith Conrad's note if you would like to see some of them. We then started on worksheet 10. Different groups worked at very different paces, but it seems like it was fairly typical to get through 10.1-10.3. We'll pick up again tomorrow.
  • Status (July 22) Great class! We proved all the lemmas, and many of the students also got a chance to derive the quartic formula.
• (July 26) Worksheet 11 -- Algebraic Elements, Minimal Polynomials
  • Status (July 26): Complete success. We solved all the problems.
•  (July 27) Worksheet 12 -- Splitting Fields
  • Status (July 27) Another complete success! The uploaded version fixes some typos from the class version.
• (July 28, July 29) Worksheet 13 -- Field homomorphisms and isomorphisms
  • Status (July 28) Slow but satisfying. People worked carefully and asked many questions. 13.1-13.3 were solved; 13.4 and 13.5 were in progress. We'll come back to this tomorrow.
  • Status (July 29) A satisfying class. Everything came together and we proved all the results.
• (August 2, August 3) Problem Set 14 -- Automorphisms of splitting fields, examples
  • Status (August 2) Groups worked at different speeds, but everyone at least thought through 14.1 - 14.3, and many details were checked. We'll resume this tomorrow.
  • Status (August 3) Everyone got to see how the key Theorem was useful, and many people saw how to use it to solve all of the problems. We'll talk a little bit more about this next time, but mostly move on.
• (August 4, August 5) Worksheet 15 -- Automorphisms of splitting fields, orbits
  • Status (August 4) We first saw how the key theorem could be used to compute the Galois group of Q(ζ) over Q, where ζ is a primitive 5-th root of unity. I had a feeling the key theorem might be hard, so I encouraged people to start with 15.3. Everyone got 15.3, but 15.4 was a real struggle, because it was one of the more abstract things we have proved, even though it was almost exactly using 13.6. We discussed it together at the end of class. Due to a internet glitch, the last few minutes weren't recorded; I have spliced an extra 5-minute video on the end of the Zoom recording to address this. We'll talk about this more tomorrow, but also move on.
• (August 5) Worksheet 16 -- a Key Short Exact Sequence
  • Status (August 5) We talked through all the parts of 16.1. Some rooms were also able to do the example problems on the rest of the worksheet; in the interest of time, we'll move on.
• (August 9, August 10)  Worksheet 17 -- Unsolvabtility of the quintic, second version
  • Status (August 9) Today involved a lot of talking by me, beacuse the main goal was to set up notation and make sure that we had formulated the theorem well. Still, we got through problems 17.1 and 17.2, as well as the first part of 17.3. We'll finish this up tomorrow. I have fixed some typos from the class version.
  • Status (August 10) We finished all the remaining problems. It turned out there were some very useful lemmas on Worksheet 14!
• (August 10, August 11) Worksheet 18 -- Solvable groups
  • Status (August 10) Problems 18.1-18.3 were solved in class; we'll finish this worksheet off tomorrow.
  • Status (August 11) We finished the worksheet and proved the Theorem! We now know that there is no formula (in terms of +, -, ×, ÷, √) for the roots of a polynomial whose splitting field has Galois group S_n for n greater than or equal to 5. We then moved to the topic of finding polynomials with such Galois groups.
• (August 11) Worksheet 19 -- Polynomials with large Galois group
  • Status (August 11) We wound up covering this topic in a lecture fashion rather than using the Worksheet, but we did establish the main results of the worksheet: Ways to make concrete polynomials whose splitting fields have Galois group S_n. Here is the Mathematica notebook from class, and here is an HTML export of it.
• (August 12) Final day! Lots of fun! Here are the slides from the final lecture

Problem Sets

• Problem set 1
• Problem set 2
• Problem set 3
• Problem set 4
• Problem set 5
• Halfway through the course, we had an opportunity to reflect on what we have done so far: Halfway Problem Set
• Problem Set 6
• Problem Set 7
• Problem Set 8
• Problem Set 9
• Problem Set 10.
• Some additional problems on Kummer theory.
• The Open Door Set