Galois Theory
Tom Leinster's Galois theory course materials from 2021 to 2023 are now on arXiv, featuring notes, videos, problems, and multiple-choice questions, gaining popularity due to enhanced preparation and visual appeal.
Read original articleTom Leinster has made his notes for the undergraduate Galois theory course he taught at Edinburgh from 2021 to 2023 publicly available on arXiv. The materials include a comprehensive account of the Galois theory content covered in the course, approximately 40 short explanatory videos, a large collection of problems, and nearly 500 multiple-choice questions. Leinster expressed surprise at the popularity of these notes compared to his previous course materials on topics like Fourier analysis and linear algebra. He attributes the success to the extra care he took in preparing the notes during the COVID-19 lockdown, as students required more guidance due to limited face-to-face interaction. Additionally, he speculated that the visual appeal of the Galois theory notes, enhanced with color and icons, may have contributed to their popularity. Leinster hopes that these resources will be beneficial and enjoyable for learners.
- Tom Leinster's Galois theory course materials are now available on arXiv.
- The resources include notes, videos, problems, and multiple-choice questions.
- The popularity of the Galois theory notes exceeds that of previous course materials.
- Leinster attributes the success to the extra care taken during the COVID-19 pandemic.
- Visual appeal may also play a role in the notes' popularity.
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- Many commenters share personal experiences of learning Galois theory, highlighting its complexity and significance in mathematics.
- Several recommend books and resources for self-study, emphasizing the importance of context in teaching mathematics.
- There is a fascination with Évariste Galois's life and contributions, noting his early death and the impact of his work.
- Some comments express a desire for simpler explanations of Galois theory for non-mathematicians.
- Common themes include the challenge of understanding abstract concepts and the need for engaging teaching methods.
He taught Galois Theory using its application to coding theory for worked examples. That class was something of a turning point in my life to be honest. I'd never think of constructing a heptagon again, for example. Definitely avoided Duels, and Montparnasse. Ok joking aside, it caused the proverbial lightbulb to turn on in my brain, and helped tremendously in my career later when I ran into folks trying to seem smart because they understood ECC or ZKPs. It was like the extreme opposite of those people who say "I never used a single thing I learned in college".
https://www.taylorfrancis.com/books/mono/10.1201/97810032139...
I've been shouting from the rooftops for years that math[0] courses need more context. We can prove X, Y, and Z, and this class will teach you that, but the motivating problem that led to our ability to do X, Y, and Z is mentioned only in passing.
We can work something out, and then come back and rework it in more generality, but then that reworking becomes a thing in and of itself. And this is great! Further advances come from doing just this. But for pedagogical purposes, stuff sticks in the human brain so much better if we teach the journey, and not just the destination. I found teaching Calculus I was able to draw in students so much more if I worked in what problems Newton was trying to solve and why. It gave them a story to follow, a reason to learn this stuff.
Kudos to the author for chapter 1 (and probably the rest, but chapter 1 is all I've had time to skim).
[0] And honestly, nearly every subject.
Start with the quadratic formula, everyone seems to have some recollection of this. Talk about solving for x in polynomials. Then discuss if you can always solve for x, and what does that even mean. If you graph a polynomial it crosses the x-axis so there's a solution for x, but does that mean you can solve for it in a formula (this alludes to the fundamental theorem of algebra that every polynomial of degree n has n solutions in the complex numbers)?
It's tough to get the idea of solution by radicals and how that relates to what it means to have a formula for x in terms of the coefficients of the polynomial.
Anyways, the punchline is that there's no formula for x using basic arithmetic operations up to taking radicals, where the formula is in terms of the coefficients of the polynomial for a general degree 5 or higher polynomial. Galois theory proves this.
Galois is credited with this because it took a lot of imagination to think about how to formulate and prove that there is no formula. What does it mean to not have a formula? How do you formulate it properly and then prove it?
Abstract interpretation models a potentially infinite set of program behaviors onto a simpler (often finite) model that is (soundly) approximate and easier to reason about (via Galois connections); here the analogy is to Galois Theory connecting infinite fields with finite groups. I often think about this when working on Value Numbering for instance.
Also (perhaps a bit of stretch) it's interesting to think of extending a computational domain (say integers) with additional values (say an error value) as a kind of field extension, and as with field extensions, sometimes (perhaps unexpectedly) complications arise (eg loss of unique factorization :: LLVM's poison & undef, or NaNs).
Relatedly, to this day I still don't know how distinguish a left-handed coordinate system from the right-handed one purely algebraically. Is the basis [(1,0,0), (0,1,0), (0,0,1)] left- or right-handed? I don't know without a picture! Does anyone?
> Flunked two colleges, fought to restore the Republic, imprisoned in the Bastille, and managed to scribble down the thoughts that would lead to several major fields of mathematics, before dying in a duel — either romantic or political — at the age of twenty.
https://www.oblomovka.com/wp/2012/09/11/touch-of-the-galois/
https://www.scribd.com/document/81010821/GaloisTheoryForBegi...
at 4:26 in https://ed-ac-uk.zoom.us/rec/play/qc1PCp8gTozfuRpMYKcTkPZQ2C...
"Do the exercises" teacher echoed over and over. I read the chapter, I followed the examples and proceed to the first problem in the unit.
My answer was 64
I go to the end of the book and the answer was 2 1/4
I would try to reverse engineer the 2 and 1/4 to original problem... Nothing!
I would ask a friend to the problem with me.. her answer was 16.
Maybe divide by 8? that gets us 2, we are closer? Right. Why divide by 8? I don't know!
Back in the there was no Internet or Kahn Academy. It was you and the red heavy book of Calculus with the desk lamp staring at you. Silently.
Direct link to PDF of notes: https://arxiv.org/pdf/2408.07499
Early morning reaction: oh god I've forgotten all of my group theory, this is bad.
After lunch: oh, right, there's only two composite numbers below 8.
Welp, guess I'm out.
He uses emacs!
OK, it's fine in some cases, but it's like a gatekeeping itself, because in order to understand Math, you need to understand Math :)
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