Name: Galois theory
ISBN: 9781032101590

What Hacker News says

susamAug 2024

Galois Theory by Ian Stewart is an excellent book indeed! I've got a hard copy lying at home that I am currently reading slowly page by page. I am planning to host book club meetups with this book later this year, perhaps during the winter if I am able to make good progress with this book.

In the meantime, if there is someone here who is interested in reading this type of books together and share updates with each other, I'd like to invite you to the IRC and Matrix channel named #bitwise [1][2] (the IRC and Matrix channels are bridged together, so you could join either one of them). The channel consists of some HN users as well as some users from other channels like ##math, ##physics, #cs, etc. It serves as an online space to share updates about mathematics and computation books you are reading and discuss their content.

[1] https://web.libera.chat/#bitwise

[2] https://app.element.io/#/room/#bitwise:matrix.org

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tobinfrickeJul 2020

Maybe a textbook like Topology by Armstrong, or Galois Theory by Ian Stewart.

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susamMay 2026

Good pedagogy is a problem even for graduate-level mathematics students and professional mathematicians. The proofs in many graduate-level mathematics textbooks are, in my humble opinion, not really proofs at all. They are closer to high-level outlines of proofs. The authors simply do not show their work. The student then has to put in an extraordinary amount of effort to understand and justify each line. Sometimes a 10-line argument in a textbook might expand into a 10-page proof if the student really wants to convince themselves that the argument works.

I am not a mathematician, but out of personal interest, I have worked with professional mathematicians in the past to help refine notes that explain certain intermediate steps in textbooks (for example, Galois Theory, by Stewart, in a specific case). I was surprised to find that it was not just me who found the intermediate steps of certain proofs obscure. Even professional mathematicians who had studied the subject for much of their lives found them obscure. It took us two days of working together to untangle a complicated argument and present it in a way that satisfied three properties: (1) correctness, (2) completeness, and (3) accessibility to a reasonably motivated student.

And I don't mean that the books merely omit basic results from elementary topics like group theory or field theory, which students typically learn in their undergraduate courses. Even if we take all the elementary results from undergraduate courses for granted, the proofs presented in graduate-level textbooks are often nowhere near a complete explanation of why the arguments work. They are high-level outlines at best. I find this hugely problematic, especially because students often learn a topic under difficult deadlines. If the exposition does not include sufficient detail, some students might never learn exactly why the proof works, because not everyone has the time to work out a 10-page proof for every 10 lines in the book.

Many good universities provide accompanying notes that expand the difficult arguments by giving rigorous proofs and adding commentary to aid intuition. I think that is a great practice. I have studied several graduate-level textbooks in the last few years and while these textbooks are a boon to the world, because textbooks that expose the subject are better than no textbooks at all, I am also disappointed by how inaccessible such material often is. If I had unlimited time, I would write accompaniments to those textbooks that provide a detailed exposition of all the arguments. But of course, I don't have unlimited time. Even so, I am thinking of at least making a start by writing accompaniment notes for some topics whose exposition quality I feel strongly about, such as s-arc transitivity of graphs, field extensions and so on.

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susamNov 2025

I learnt this subject from the book Galois Theory, 5th ed. by Ian Stewart. Quoting from page 177:

Theorem 15.10. The polynomial t⁵ - 6t + 3 over ℚ is not soluble by radicals.

As you can see, this theorem occurs in Chapter 15. So it takes fourteen chapters before we reach here. It takes a fair amount of groundwork to reach the point where the insolubility of a specific quintic feels natural rather than mysterious.

To achieve this result, the book takes us through a fascinating journey involving field extensions, field homomorphisms, impossibility proofs for ruler and compass constructions, the Galois correspondence, etc. For me, the impossibility proofs were the most interesting sections of the book. Before reading the book, I had no idea how one could even formalise questions about what is achievable with a ruler and compass, let alone prove impossibility. Chapter 7 explains this beautifully and the algebraic framework that makes those proofs possible is very elegant.

By the time we reach the section about the insoluble quintic, two key results have been established:

Corollary 14.8. The symmetric group S_n is not soluble for n ≥ 5.

Theorem 15.8. Let f be a polynomial over a subfield K of ℂ. If f is soluble by radicals, then the Galois group of f over K is soluble.

The final step is then quite neat. We show that the Galois group of f = t⁵ - 6t + 3 over ℚ is S₅. Corollary 14.8 tells us S₅ is not soluble. By the contrapositive of Theorem 15.8, f is not soluble by radicals.

Obviously whatever I've written here compresses a huge volume of work into a short comment, so it cannot capture how fascinating this subject is and how all the pieces fit together. But I'll say that the book is absolutely wonderful and I would highly recommend it to anyone interested in the subject. The table of contents is available here if you want to take a look: https://books.google.co.uk/books?id=OjZ9EAAAQBAJ&pg=PT4

Two small warnings: The book contains a fair number of errors which can be confusing at times, though there are plenty of errata and clarifications available online. And unless you already have sufficient background in field homomorphisms and field extensions, it can take several months of your life before you reach the proof of the insoluble quintic.

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susamApr 2025

What a coincidence! Just a few minutes ago, I finished reading Chapter 20 ("Regular Polygons") from the book Galois Theory, 5th ed. by Ian Stewart. It presents a rigorous proof of why the regular 65537-gon can be constructed with an unmarked ruler and compass. In fact, the book proves a more general result.

Firstly, the regular 65537-gon can be constructed using only an unmarked ruler and compass because 65537 is a Fermat prime, i.e., a prime of the form, 2^(2^r) + 1, where r is a non-negative integer. Indeed 65536 = 2^(2^4) + 1.

The more general result can be stated as follows: The regular n-gon can be constructed by an unmarked ruler and a compass if and only if n has the form

  n = 2^r p_1 ... p_s

where the integers r, s >= 0 and p_1, ..., p_s are distinct Fermat primes. As a result, if n is a Fermat prime, the regular n-gon is constructible. This also explains why, for example, the regular 7-gon, 9-gon, etc. are not constructible by unmarked ruler and compass.

Remarkably, the only Fermat primes known so far are 3, 5, 17, 257, 65537. See also <https://oeis.org/A019434>.

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susamJul 2024

My own form of learning in public has been to host book clubs for the books I have been reading and then taking notes, sharing insights about them through blog posts, discussing them on IRC channels, etc. I have done two such book clubs so far. The first one was for Introduction to Analytic Number Theory (Apostol, 1976) and the second one was for Mastering Emacs (Petersen, 2022).

I have the meeting logs, details, etc. for them available here: https://susam.net/cc/

I plan to begin one more on Galois Theory (Stewart, 2015) quite soon. I am just waiting for the hardcover book to arrive! If this type of thing interests you, I'd love for you to join the discussion either in the IRC channel or the Matrix room linked on the page above.

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