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August 15, 2024

Galois Theory

Posted by Tom Leinster

I’ve just arXived my notes for Edinburgh’s undergraduate Galois theory course, which I taught from 2021 to 2023.

I first shared the notes on my website some time ago. But it took me a while to arXiv them, because I wanted to simultaneously make public most of the other course materials. I now have, which means the following are now available to all:

  • Notes forming a complete, self-contained account of the part of Galois theory that we covered.

  • About 40 short explanatory videos.

  • A large collection of problems.

  • Nearly 500 multiple choice questions.

I’m a little bemused by the popularity of the Galois theory notes. I’ve made quite a few sets of course notes public before, e.g.:

But the Galois theory notes seem to have caught on in a way that none of the others have (except category theory — but that one, I made into a book).

It’s true that I probably took a bit of extra care on them: I first taught the course during full Covid lockdown, and I felt the students would need more guidance than usual, given that they were deprived of all opportunities for face-to-face interaction. But I wonder whether the real reason is that the Galois theory notes simply look nicer, with colour and little icons and so on.

In any case, I hope the notes, videos and questions bring people joy.

Posted at August 15, 2024 1:02 PM UTC

TrackBack URL for this Entry:   https://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/3551



17 Comments & 0 Trackbacks

Re: Galois Theory

thank you for sharing these publicly!

Posted by: gnv on August 15, 2024 3:47 PM | Permalink | Reply to this

Re: Galois Theory

From my biased perspective, the popularity is because Galois theory is simply prettier than almost all the rest of undergraduate math (with stiff competition from complex analysis and rep theory of finite groups).

Posted by: Asvin on August 15, 2024 5:07 PM | Permalink | Reply to this

Re: Galois Theory

You’re absolutely right; I shouldn’t have overlooked that factor! Galois theory is such an amazing story.

Posted by: Tom Leinster on August 15, 2024 5:34 PM | Permalink | Reply to this

Re: Galois Theory

The hackers are interested..

Posted by: Simon Burton on August 15, 2024 6:38 PM | Permalink | Reply to this

Re: Galois Theory

the popularity has to do with the cachet the name brings. If it was called Smith Theory or even Leinster Theory, nobody would care. But Galois Theory - please tell me more!

Posted by: ag on August 15, 2024 11:32 PM | Permalink | Reply to this

Mersenne Twister

I was looking for a good explanation of the Mersenne Twister algorithm.

For example, there are magic constants such as a=9908B0DF (base 16). How would you calculate these if you didn’t know them already?

I tried reading the paper, but was only able to understand about half of it.

This is related to Galois Theory, because Merseene Twister is multiplication in a finite field of 2^p elements, where p is a Mersenne exponent.

Posted by: FSK on August 16, 2024 12:11 AM | Permalink | Reply to this

Re: Galois Theory

Thank you so much for sharing so many materials! And I really love also your other sets of course notes!

Posted by: Paolo on August 16, 2024 1:07 AM | Permalink | Reply to this

Re: Galois Theory

yay for Galois theory

Posted by: a on August 16, 2024 3:57 AM | Permalink | Reply to this

Re: Galois Theory

Great! I’ll add this to my list of resources for how to learn math.

How in the world did you have the energy to write 477 multiple choice questions? Was that required because you taught many sections of the class and needed to keep creating new questions?

Posted by: John Baez on August 16, 2024 4:18 PM | Permalink | Reply to this

Re: Galois Theory

Thanks!

Here’s the story. I first taught the course in spring 2021, which in Edinburgh was the second fully locked down semester of teaching. What this meant was that by the time I came to design my course, we’d already had plenty of feedback from the students about the challenges of learning in lockdown. One point that came out strongly was that many students found it hard to know whether they were making adequate progress. I guess attending class in person and being around your classmates gives you reassurance that you’re doing OK, and the absence of that was leading to insecurity.

To try to do something about this problem, I came up with the idea of a random multiple choice quiz that you could take any time, with ten random questions each time you took it. It was called the “How am I doing?” quiz. The “I” refers to the student, not me — it’s not like “How’s my driving?” :-)

As for how I had the energy to write 477, they just came pouring out. Most of them are true/false statements, which are easy enough to come up with. With ten weeks in the semester, that’s only 40–50 per week. I didn’t set myself a goal of how many to write, and there was only one section of the class. That was just how many I thought of. I found it quite pleasant, and infinitely less work than writing the notes or recording the videos.

Posted by: Tom Leinster on August 17, 2024 3:41 PM | Permalink | Reply to this

Re: Galois Theory

I came from hacker news site where I found this post. You pique my interest in this theory so thank you.

Posted by: Younes Ben Amara on August 17, 2024 6:41 PM | Permalink | Reply to this

Re: Galois Theory

Thanks! Glad to have piqued your interest.

Posted by: Tom Leinster on August 18, 2024 11:41 AM | Permalink | Reply to this

Re: Galois Theory

“I’m a little bemused by the popularity of the Galois theory notes. […] But the Galois theory notes seem to have caught on in a way that none of the others have”

I studied at a different university, in a different decade, and haven’t gotten the chance to read your notes yet. Perhaps the answer is as simple as: Galois theory is one of the most fun parts of mathematics — and linear algebra one of the least!

I never understood why schools teach math in the order they do. (Probably for the engineers.) I would have been seriously turned off of the field entirely, had I been unlucky enough to follow the standard path. “Congrats on passing calculus! Now here’s a year of math that will make you think you registered for a 200-level course in Watching Paint Dry.”

Posted by: K on September 3, 2024 2:10 PM | Permalink | Reply to this

Re: Galois Theory

Right, agreed! See my reply to Asvin above. Galois theory is a hugely appealing subject.

Personally, as an undergraduate I loved linear algebra. Well, I didn’t much like the first-year course on it (a mishmash of matrix methods and linear transformations), but the second-year course on the abstract theory of finite-dimensional vector spaces lit up my world. We were lucky enough to have a lecturer who didn’t hold back from letting the abstract beauty shine through. For example, he went carefully through the theory of dual vector spaces, and even now I can’t put into words how supremely satisfying I found it.

The lecturer in question was the analytic number theorist Roger Heath-Brown, and I guess analytic number theorists aren’t particularly renowned for being attuned to the beauty of abstract algebraic structures, but that course was pure joy.

Posted by: Tom Leinster on September 3, 2024 5:45 PM | Permalink | Reply to this

Re: Galois Theory

Fantastic videos, Tom, congratulations!

I have a left-field question, feel free to ignore of course. When Galois himself described the underlying symmetry groups, his own criterion for ‘solvability’ was whether each group contained a prime number of elements. I assume this must have turned out to be wrong, as I have never heard it referred to anywhere else. Thanks.

https://mathshistory.st-andrews.ac.uk/Projects/Brunk/chapter-3/

Posted by: Bertie on September 23, 2024 8:38 AM | Permalink | Reply to this

Re: Galois Theory

I should have added that by far the most likely explanation is that I misunderstood his writings! They aren’t especially easy to follow?

Posted by: Bertie on September 23, 2024 9:39 AM | Permalink | Reply to this

Re: Galois Theory

I hadn’t heard that! That’s weird.

Posted by: Tom Leinster on September 23, 2024 12:11 PM | Permalink | Reply to this

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