[HN Gopher] Lemma for the Fundamental Theorem of Galois Theory
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Lemma for the Fundamental Theorem of Galois Theory
Author : susam
Score : 89 points
Date : 2025-03-15 15:38 UTC (7 hours ago)
(HTM) web link (susam.net)
(TXT) w3m dump (susam.net)
| sd9 wrote:
| Galois theory was my favourite course in the final year of my
| maths undergrad.
|
| I have no idea how this post is at the top of HN. I barely
| remember what the symbols mean.
| dboreham wrote:
| Similarly, Galois theory was a penny dropping moment in my
| understanding of mathematics and has remained surprisingly
| useful late into by career, as software has become increasingly
| "mathy".
| Bayes7 wrote:
| Would you mind sharing how it turned out useful?
| frabcus wrote:
| Me too! But I can't really remember why. The proof was
| impressive.
| JadeNB wrote:
| Galois theory is so useful that, besides its fundamental
| importance in algebra, it birthed the whole subject of Galois
| connections, which crop up all over the place, including in
| theoretical CS: https://en.wikipedia.org/wiki/Galois_connection
| .
| nextos wrote:
| Galois connections are super useful for static program
| analysis.
|
| In particular, for abstract interpretation. A great intro
| book is [1].
|
| [1] Program Analysis - An Appetizer.
| https://arxiv.org/pdf/2012.10086
| RheingoldRiver wrote:
| As another math major who doesn't do math anymore, I feel this
| comment so deeply in my soul
| susam wrote:
| Hello HN! Thank you for upvoting this post to the front page. I'm
| quite delighted to see this post about a relatively obscure lemma
| from Galois theory make it to the front page.
|
| This post is primarily intended as notes on the subject. The
| theorems and proofs involved in Galois theory can often feel too
| abstract, so I find it helpful to work through them with concrete
| examples. This is one such example that I wanted to archive for
| future reference on my website. If you spot any errors, please
| let me know.
| JadeNB wrote:
| Thank you for sharing your notes--everyone needs a different
| push/viewpoint to make key concepts click, and, the more
| expositions are out there, the more likely someone will be to
| find the one that's right for them!
|
| Is M^* Stewart's notation? I'm much more used to seeing it used
| for a dual space than a group of automorphisms. Conventions
| aside, it's a bit unfortunate in that it doesn't specify the
| field on which we're acting! I think that the notation
| Aut(L/M), or Gal(L/M) for a Galois extension, is more common.
| susam wrote:
| Thank you! Yes, M^* is Stewart's notation. Indeed this
| notation is "lossy" in the sense that it loses information
| about what the extension field is. The extension field is
| usually introduced once for every example or theorem and then
| M^* is used as a shorthand notation. For example, in my post,
| the extension field is introduced like this:
|
| _" The notation M^* denotes the group of all M-automorphisms
| of L with composition as the group operation."_
|
| By the way, Stewart uses the notation G(L/M) too at several
| places but in this specific lemma, the notation M^* is used
| and therefore my post too uses the same notation.
| enriquto wrote:
| The html source is beautiful, did you write it by hand?
| susam wrote:
| Thanks! Yes, I handcraft all my HTML and CSS. I'm glad you
| noticed the HTML and liked it. I find great joy in crafting
| my website by hand. It's like digital gardening. I grow all
| my HTML and CSS myself. It's all 100% organic and locally
| sourced!
| enriquto wrote:
| I rarely ^U nowadays, but your site was so clean that I
| couldn't resist!
|
| Just as a side note : when writing html5 by hand, you can
| use the full power of the language, most notably optional
| tags (no need to write html, body, etc) and auto-closing
| tags (no need to close p, li, td, etc). You may get
| something even crispier!
|
| See for a reference the google html style guide : https://g
| oogle.github.io/styleguide/htmlcssguide.html#Option...
|
| And the official html5 reference for a complete list of
| optional tags :
| https://html.spec.whatwg.org/multipage/syntax.html#syntax-
| ta...
| susam wrote:
| > Notice that, when writing html5 by hand, you can use
| the full power of the language, most notably optional
| tags (no need to write html, body, etc) and auto-closing
| tags (no need to close p, li, td, etc). You may get
| something even crispier!
|
| Yes! In fact, sometime back I wrote a little demo page to
| show the minimal (but not code-golfed) HTML we can write
| such that it passes validation both with the Nu HTML
| Checker and HTML Tidy.
|
| Here's the demo page:
| https://susam.net/code/web/minimal.html
|
| Here's the Nu HTML Checker output: https://validator.w3.o
| rg/nu/?doc=https%3A%2F%2Fsusam.net%2Fc...
|
| Here's the HTML Tidy (version 5.8.0) output:
| $ tidy -quiet -errors minimal.html $
|
| Here's the HTML: <!DOCTYPE html>
| <html lang="en"> <meta charset="UTF-8">
| <title>Hello</title> <body> <p>Hello, World!
|
| That said, when writing my own posts, I prefer keeping
| optional and closing tags intact. Since I use Emacs, I
| can insert and indent closing tags effortlessly with C-c
| /. It's a bit like how some people write:
| 10 PRINT"HELLO
|
| But I've always preferred: 10 PRINT
| "HELLO"
|
| I find the extra structure more aesthetically pleasing.
| zyklu5 wrote:
| Let me take this opportunity to post one of the best texts on
| Galois Theory I have read -- and I had to go through quite a few
| while preparing for a class.
|
| https://pages.uoregon.edu/koch/Galois.pdf
|
| The subject is developed very naturally and every idea is
| beautifully motivated. It begins with a quick one chapter intro
| of Arnold's proof of Abel-Ruffini.
|
| Richard Koch's home page (https://pages.uoregon.edu/koch/) has
| other examples of his fantastic pedagogy.
| almostgotcaught wrote:
| > It begins with a quick one chapter intro of Arnold's proof of
| Abel-Ruffini.
|
| The key to understanding/motivating Galois theory is Abel-
| Ruffini, which is a corollary of Galois. And the simplest way
| to understand _that_ is Arnold 's topological proof, which i
| learned about from this video
|
| https://www.youtube.com/watch?v=RhpVSV6iCko
|
| Watching that video and rolling it around in my head completely
| demystified Galois theory for me, _years_ after literally 2
| semesters of algebra in undergrad. Everything about normal
| subgroups and commutators and splitting fields and blah blah
| blah immediately became tangible and obvious. It should be a
| crime not teach this proof first.
|
| The coverage in Koch's book looks good too - lots of pictures -
| and funny enough it links to a different youtube video.
|
| Edit: copy-pasting notes I took from the video after watching.
|
| ------
|
| The idea is to continuously perturb each of the coefficients of
| the polynomial along a loop (change each of them from their
| initial value such that they traverse a path that returns them
| to that initial value at the end of the path) and study what
| happens to the roots of the polynomial.
|
| Note, once all coefficients have returned to their original
| values the entire set of roots also returns to itself, but each
| root does not necessarily returns to its original value. In
| general you get a permutation of the set of roots and so in
| this way we get a mapping between loops of the coefficients and
| permutations of the roots.
|
| Also note, we can produce coefficient loops that map to any
| permutation of the roots by permutating the roots and
| "watching" the coefficients.
|
| Hence, the way to prove Abel-Ruffini is to show that any
| expression involving the coefficients (ie formula for the roots
| in terms of the coefficients) returns to itself after the
| coefficients traverse their loops _but the roots do not_ (and
| therefore the expression cannot capture all of the roots). For
| example, an immediate corollary of the construction of the
| mapping between loops of coefficients and roots is the fact
| that a general solution involving only -, +, x, / is not
| possible; -, +, x, / are all single-valued and therefore no
| composition thereof could produce multiple roots.
| zyklu5 wrote:
| There is indeed a deep connection between what is going on
| behind Arnold's proof and the classical Galois theory. But it
| needs quite a bit of sophistication to flesh out properly
| (not apparent in his famous lectures given to high school
| kids). There is a Galois theory for Riemann surfaces over
| algebraic functions where the coverings behave like fields do
| in the classical correspondence. If any one is interested,
| check out chapter 3 of Khovanskii's Galois Theory, Coverings
| and Riemann Surfaces.
| almostgotcaught wrote:
| i mean calling arnold's proof actually topological is
| probably a stretch (the "deep connection" you're talking
| about). it doesn't really use any topological facts about
| either the loops or the embedding space. continuity isn't
| really required i don't think? i should've said contiguity
| instead. it's just a very very nice model for the theory
| (in the sense of model theory) that lends itself to
| immediate visualization.
| galaxyLogic wrote:
| Is there such a thing as the graph of lemmas and theorems in
| mathematics, somewhere online?
|
| I can't really understand much of mathematics, but I would
| appreciate seeing the graph of how all major proofs are
| constructed with help of other proofs.
| xanderlewis wrote:
| You mean like a dependency tree? There isn't a comprehensive
| one -- it would be way too big and we don't have all
| mathematical knowledge written down in one place anyway.
|
| But if you're interested in any given classical result, if you
| ask I'm sure a mathematician can give you a rough idea of what
| the path back down to first principles is. If you're not
| literally interested in starting by assuming the existence of
| the empty set, say, then opening any introductory book on the
| topic will give you an idea. Just look at the proof of the
| result and follow its references back. It won't go _that_ deep.
| galaxyLogic wrote:
| I see. I was just thinking I would personally appreciate
| seeing a big graph of proof-dependencies. A bit like "Seeing
| the Forest from the Trees".
|
| Every mathematician is working on their own TREE, I assume.
| But might be useful if somebody was looking at how the FOREST
| is doing.
| jeremyscanvic wrote:
| The thing with mathematics is there are often many different
| ways to prove things leaving no universally agreed upon graph
| of true mathematical statements in terms of which helps proving
| which! That being said, you might want to check out ProofWiki
| [1], an online collaborative project featuring mathematical
| proofs consistently linked together.
|
| [1] https://proofwiki.org/wiki/Main_Page
| galaxyLogic wrote:
| Ah, just what I was asking for. Thanks
| reuben364 wrote:
| There is a large library of mathematics formalized in lean
| called mathlib. There are graphs for the module dependencies,
| but I haven't seen any at the level of definitions (including
| lemmas and theorems).
| jurgenaut23 wrote:
| It is quite fascinating to think that Galois lived only 20 years
| in the early 19th century, yet he conceived a theory of profound
| significance and impact 200 years later. Imagine if he could live
| 80 years!
| xanderlewis wrote:
| I'm inclined to agree, but who knows... maybe he just happened
| to have his big insight early on, and if he'd lived longer he'd
| never have done anything quite as significant. Plenty of people
| do great things only once.
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