22 ms·

Galois Theory

467 points by mathgenius 2y ago

082349872349872 2y ago
In particular, abuse of Galois Theory makes it possible to reconcile Spinoza with Aquinas.
- esafak 2y ago
  Go on.
  - koolala 2y ago
    Reconcile Liebniz too. I agree this sounds awesome.
    - 082349872349872 2y ago
      Leibniz does get a mention, but it'll already take a while to explain these two, so I'll leave his full reconciliation as an exercise for the reader ;)
      - koolala 2y ago
        ;) A mention is enough. Monads are LIFE!
        082349872349872 2y ago
        Every mathematical monad comes from an adjunction, which unfortunately implies they're not as windowless as Leibniz would like? (or does it merely imply that in the Leibnizian setting the structure is always external and never internal?) Can we make a mathematical monad out of the adjunction presented above? Let g(C) be the minimum G and c(G) be the maximum C in the model above, then we certainly have g(C) = g(c(g(C))) and c(G) = c(g(c(G))), which formally suggest we may be able to do something. Exercise: does the triple (T, μ, η) exist? what about (G, δ, ϵ) in the other direction? If they do exist, what are they, for Aquinas, Leibniz, and Spinoza?
- novosel 2y ago
  Why would they need to be reconciled? Or, indeed, Galoas abused?
  - 082349872349872 2y ago
    They don't need to be reconciled, but they do differ, which suggests we might be able to find a framework which reconciles them. Galois Spinoza Aquinas in Google yields (for me at least) the following HN thread: https://news.ycombinator.com/item?id=39885475 https://news.ycombinator.com/item?id=39885475 , in which an attempt at "Algebraic Theology" not only provides such a subsuming (thanks to Galois Theory) framework, answering that particular question in the affirmative, but also raises many other questions which might be amusing to pursue. (I could summarise those Q's in this thread, if any of you all are more interested than the downvoters were) [I will summarise that thread further up in this one, but as it was months ago it may take me an hour or two to page everything back in]
- 082349872349872 2y ago
  To expand[0] upon that claim: Aquinas, Leibniz, and Spinoza all agree that there is a God (G) that created[1] a Creation (C) which we are a part of. One major way Aquinas (and Leibniz) differ from Spinoza is in how determinate C may be. Spinoza says C is determined by G; Aquinas says there are many possible C's for any given G. (Leibniz splits the difference and says there are many possible C's, but in our particular case, our G has created the best[2] possible C.) The reconciliation: let G and C be in an adjoint relationship, such that we have functions picking out the maximum C any given G may create, and the minimum G that can create any given C. Now, if you are Aquinas, G is omnipotent, and hence has the possibility to create other C's, but our C is the maximum[3] one. On the other hand, if you are Spinoza, G determines C[4], so it is trivially maximal. (The maximum of a singleton being the unique element) Does that make sense? Question: do there exist Gods that are incapable of creating any creation, or Creations that are impossible for any god to create? If so, need we replace "maximum" and "minimum" above by LUB and GLB? What other situations (eg. gods or creations being only domains rather than lattices) would also require further abstraction? :: :: :: [0] my apologies for any non-standard notation. I tried to find a survey paper on "Algebraic Theology" so I could follow the existing notation, but failed to find any concrete instances in this (currently only platonic?) field. [1] Spinoza is accused of "pantheism": the heresy of identifying God and the Creation. Reading him according to a Galois-theoretic model, he would be innocent of this heresy, for when he says "God, or, the Universe", he is simply using metonymy, for in his model God and Creation are dual, so (being in a 1:1 relationship) one determines the other. Note that in general, not only are duals not identical, they're not even isomorphic. [2] but cf Voltaire, Candide (1759) [3] if you are Leibniz, the order in which it is maximal corresponds to the traditional "worst" "better" "best" order. I don't know Leibniz well enough to say if he had a total, or only partial, order in mind; presumably in his model if there are several maximal "best" creations they would all be isomorphic? Exercise for the reader: work out the Leibnizian metaphysical adjunction. [4] turning this arrow around, C determines G, which explains why Einstein would say he believed in the "God of Spinoza", and chose to base his research by thinking about C, unlike the medieval colleagues of Aquinas, who spent a lot of time and effort trying to work the arrow in the other direction, hoping to come to conclusions about C in starting by thinking about G.
  - 2y ago
rpmw 2y ago
I will always remember Galois theory as the punchline to my Abstract Algebra courses in college. Galois was a brilliant math mind, and I'm curious what else he would have contributed had he not died at 20 in a duel. https://en.wikipedia.org/wiki/%C3%89variste_Galois https://en.wikipedia.org/wiki/%C3%89variste_Galois
- mcfunley 2y ago
  Great to see this course material public. It's a real missed opportunity though to not mention that Galois wrote a lot of it down staying up all night before being shot.
  - bubble12345 2y ago
    That's a common myth. See this paper referenced in the wikipedia article: Rothman, Tony (1982). "Genius and Biographers: The Fictionalization of Evariste Galois". The American Mathematical Monthly. 89 (2): 84–106. doi:10.2307/2320923. JSTOR 2320923
    - ForOldHack 2y ago
      Quite the clickbait. You can only access it from the pay site, or unless you can get a school library to access it, which I will do. Only the first page is available free.
      - bubble12345 2y ago
        Or you can paste the following JSTOR link into sci-hub. https://www.jstor.org/stable/2320923 https://www.jstor.org/stable/2320923
- agumonkey 2y ago
  I'd be genuinely curious to have a chat with similar minded people. They must see the world quite differently to be able to fork a new path in hard maths mostly on their own ..
frakt0x90 2y ago
My second semester of algebra had a section on Galois theory and I remember thinking it was abstract nonsense and I didn't get it. I'm actually interested in going through this to see if my perspective has changed.
- openasocket 2y ago
  It's all abstract nonsense until it starts having practical results lol :) The main motivation for Galois theory was proving the insolvability of the quintic. For those not aware, there is a general formula solving the quadratic equation (i.e. solving ax^2 + bx + c = 0). That formula has been known for millenia. With effort, mathematicians found a formula for the cubic (i.e. solving ax^3 + bx^2 + cx + d = 0), and even the quartic (order 4 polynomials). But no one was able to come up with a closed-form solution to the quintic. Galois and Abel eventually proved that a quntic formula DOES NOT EXIST. At least, it cannot be expressed in terms of addition, subtraction, multiplication, division, exponents and roots. You can even identify specific equations that have roots that cannot be expressed in those forms, for example x^5 - x + 1. I took an entire course on it that went through the proof. It's actually very interesting. It sets up this deep correspondence between groups and fields, to the point that any theory about groups can be translated into a theory about fields and vice versa. And it provides this extremely powerful set of tools for analyzing symmetries. The actual proof is actually really anticlimactic. You have all these deep proofs about the structures of roots of polynomial equations, and at the very end you just see that the structures of symmetries of certain polynomials of degree 5 (like x^5 - x + 1) don't follow the same symmetries that the elementary mathematical operations have. Literally, the field of solutions to that polynomial doesn't map to a solvable group: https://en.wikipedia.org/wiki/Solvable_group https://en.wikipedia.org/wiki/Solvable_group In almost the same breadth, it also proves that it is impossible to trisect an angle using just a compass and straightedge, a problem that had been puzzling mathematicians for millennia. It's actually almost disappointing: we spent then entire course just defining groups and fields and field extensions and all this other "abstract nonsense". And once all of those definitions are out of the way, the proof of the insolvability of the quintic takes 10 minutes, same for the proof of impossibility of trisecting an angle.
  - 2y ago
VyseofArcadia 2y ago
Chapter 1 is brilliant. 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.
- gowld 2y ago
  Hot take: "18th Century" mathematics was intiuitive and informal, to the point that it was inconsistent. The 19th and 20th Centuries added rigor and formalism (and elitism) and devalued intuition, to the point that it begame uninterpretable to most. The 21st Century's major contribution to mathematics (including YouTube! and conversational style writing) was to bring back intuition, with the backing of formal foundations.
  - Agingcoder 2y ago
    I like your idea. I was schooled in abstract 20th century math - indeed YouTube is the opposite, and it’s a good thing. One of my math teachers was once talking to Jean Dieudonné https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9 https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9 who was part of the Bourbaki group and asked him why on earth he insisted on inflicting raw dry theory to the world with no intuition , when his day job involved drawing ideas all day long ! Edit: interestingly enough, one of my colleagues thinks very strongly that intuition should not be shared, and the path to intuition should be walked by everyone so that they ´ Make their own mental images ´. I guess that there’s a tradeoff between making things accessible, and deeply understood, but I don’t know what to make of his opinion.
gowld 2y ago
Notes, Videos, and Problems: https://www.maths.ed.ac.uk/~tl/galois/#notes https://www.maths.ed.ac.uk/~tl/galois/#notes Direct link to PDF of notes: https://arxiv.org/pdf/2408.07499 https://arxiv.org/pdf/2408.07499
mkw5053 2y ago
A few years ago, I led a study group through A Book of Abstract Algebra by Charles C Pinter. It culminated in Galois Theory and was one of the best books I've ever used in a math study group.
- mkw5053 2y ago
  Also, it assumes little to no advanced math knowledge and can be found for free online :)
- sifar 2y ago
  I second this book, really accessible. It taught me Abstract Algebra when I was learning it by myself.
- bosquefrio 2y ago
  I have fond memories of reading this book in college. I enjoyed it immensely. I also remember reading the section at the back of the book about Galois. There was also an entertaining section about the history of solving the roots of polynomial equations and in particular solving equations of arbitrary order.
PreInternet01 2y ago
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- ogogmad 2y ago
  I don't think this is Galois theory. Galois theory is about "The Fundamental Theorem of Galois Theory", which states that there is a nice mapping from field-extensions to group-extensions, where the resulting groups are usually finite. When the resulting groups are finite (as they usually are), many problems involving field extensions can be solved using brute-force search. Galois fields happen to be something else named in honour of Galois.
  - koolala 2y ago
    Galois Fields yeah. Being able to store Rational Numbers in a computer. "Originally, the theory had been developed for algebraic equations whose coefficients are rational numbers."
    - matt-noonan 2y ago
      That is definitely not what Galois Fields are about.
      - koolala 2y ago
        https://en.m.wikipedia.org/wiki/Galois_theory#Permutation_group_approach https://en.m.wikipedia.org/wiki/Galois_theory#Permutation_gr... That quote about equations with rational numbers was from here. Galois didn't have a computer of course. Rational Numbers and trisecting an angle sound related.
        wging 2y ago
        The quote is followed immediately by this: "It extends naturally to equations with coefficients *in any field*, but this will not be considered in the simple examples below." Emphasis on 'in any field' is mine. Among the other fields that can be considered include the Galois fields, which are another name for finite fields. (There are also infinite fields other than the rationals, so 'in any field' does not just mean Galois fields/finite fields.) https://en.wikipedia.org/wiki/Finite_field https://en.wikipedia.org/wiki/Finite_field Galois fields have nothing to do with being able to represent rational numbers in a computer: elements of a finite field aren't even rational numbers.
ogogmad 2y ago
Interestingly, there's a close connection between the "Fundamental Theorem of Galois Theory" and the "Fundamental Theorem of Covering Spaces".
HPsquared 2y ago
For non-math people, is this "simple Wikipedia" article about right? I've always seen Galois theory listed in mathematics courses and wondered what it is, speaking as a humble engineer. https://simple.m.wikipedia.org/wiki/Galois_theory https://simple.m.wikipedia.org/wiki/Galois_theory
- gnulinux 2y ago
  Yes and no, it's a bit too simplistic and doesn't explain the actual "why" of Galois Theory, just the how. The brilliant insight Galois figured out is that there is a fundamental connection between fields and groups, but that is just the "technique" with which he solved the problem. The "why even bother" is a bit more complex but simply put Galois wanted to establish a criterion to determine what polynomials are solvable or unsolvable in which fields. E.g. we know x^2=-1 is solvable in C with x=i but not in real numbers. Can we generalize that proof to such a degree that we can mechanistically run it for arbitrary polynomials in arbitrary fields?
- fredilo 2y ago
  What it states is correct and it gives you a good overview over what you do in a Galois theory course. It does, however, not give you an idea of why this is interesting. When just reading that article one might get the idea that some mathematicians just had too much free time. I tried to motivate the questions leading to Galois Theory in https://news.ycombinator.com/item?id=41258726 https://news.ycombinator.com/item?id=41258726 in a way that is hopefully accessible to more down-to-earth programmers and engineers.
  - fredilo 2y ago
    I should probably add why I think the motivation is so important here. For pure engineers, numbers are a tool. They ask: What can I build with numbers? Pure mathematicians ask a different question. They are interested in the limits of numbers. They ask: What can I not build with numbers? Studying these two questions is deeply related but also a constant source of frustration for engineers taking math courses designed for mathematicians by mathematicians. Galois theory, is a theory of "no". It ultimately serves to answer several "Can I build this?" questions with no. This makes it very interesting to pure mathematicians. However, for pure engineers that are looking for numeric machine parts that can be assembled in other useful ways to actually build something... Galois theory can be quite disappointing.
raldi 2y ago
Clicked around for a few minutes and couldn’t find a sentence beginning, “Galois Theory is…”
- chii 2y ago
  https://www.maths.ed.ac.uk/~tl/galois/ https://www.maths.ed.ac.uk/~tl/galois/ it's a link to the course, and there's an introduction right at the beginning of the course. Unfortunately, it's a video, rather than text, but there's at least transcript. Even wikipedia acknowledges that it is not a simple subject: https://simple.wikipedia.org/wiki/Galois_theory#Disclaimer https://simple.wikipedia.org/wiki/Galois_theory#Disclaimer ;so hoping you can get an understanding of it in a few sentences is probably asking too much tbh.
- skulk 2y ago
  Someone could tell you that Galois theory is the study of field extensions and the structure of their automorphism groups but is that really going to help?
  - gowld 2y ago
    The fundamental theorem of Galois theory reduces certain problems in field theory (like finding roots of polynomials) to group theory (counting permutations and symmetries), which makes them simpler and easier to understand and solve (or prove non-solvable). Galois theory is the proof and applications of this theorem, and related topics. Key to this theory is being more methodical about extending the rational numbers into the real numbers, by introducing new numbers one at a time, instead of all at once. One of the immediate discoveries in beginning this study is the fact that in many common cases you cannot add just one numbers one at a time, but must add 2 or more numbers at once. These sets of numbers are called conjugates, and have the interesting property that even though you can prove how many must exist and that they are distinct from each other, they are otherwise identical except in the arbitray names you give them.
- taneliv 2y ago
  I watched one of the videos, titled "Galois groups, intuitively".[1] It is about 18 minutes long and gave me a somewhat understandable overview what Galois groups are (not the theory yet!). I suppose if you have the time to spare, at least it should give you an idea if the topic is of interest and whether you need to remind yourself of some mathematical concepts to be able to study rest of the material, or if it is too elementary for you and a shorter treatise from elsewhere would be preferable. (I actually thought I had learned something about Galois theory in University algebra, but either I hadn't, or I've forgotten more than I wanted to admit. Which is to say, if you watch the video, your mileage may vary!) [1] https://ed-ac-uk.zoom.us/rec/share/_I-EeZA8_399ArdZ1GyKtM_rDWHI8uE_n4YysAZwJA1EF3Ix2cgRkxoABrZEU0td.sSYj18b7bczdplw0?startTime=1610319766000 https://ed-ac-uk.zoom.us/rec/share/_I-EeZA8_399ArdZ1GyKtM_rD... (link copied from the page, I hope it works)
Joker_vD 2y ago
> But then you realize something genuinely weird: There’s nothing you can do to distinguish i from −i. 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?
- solveit 2y ago
  Stack the vectors up so it's a matrix and take the determinant. The sign tells you which one it is.
  - gowld 2y ago
    No, it does not. The determinant tells you whether two bases have the same or different handedness, not which one is "left" or "right".
    - lanstin 2y ago
      It's 2 cosets, one is arbitrarily left handed and the other arbitrarily right handed. If you are in an orientable space :) if not, then there's no global concept of left or right. a
      - fredilo 2y ago
        Formulated differently, you cannot determine left- and right-handedness but you can determine same-handedness.
- gowld 2y ago
  No, it's impossible, even in principle, to answer that question. You can draw a picture for either answer. "Left" and "right" are a dipole. Neither one can exist without the other, and they are symmetric. It's the same issue as we have with the conjugates discussed in Galois Theory. In fact, in an algebraic (non-ordered/arithmetic/analytic) perspective, it's misleading to use the symbols + and - to label the conjugates in field extensions like sqrt2 and i. Left and Right are better names than + and - for those conjugate pairs. Only when we impose an arithmetic ordering (which is not needed in the theory of algebraic equalities) is it meaningful to use + and -: -sqrt(x) < 0 < +sqrt(x), where x is a positive real number. (and when x is a negative real number, we immediately see the problem with - again: -i and i are not separable via ordering with respect to 0.)
revskill 2y ago
The problem with many mathematics books, is it uses Math to teach Math !!! 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 :)
- koolala 2y ago
  Painting, Sculpture, Music, Art is a Language. We need the Toki Pona of Math. I hope Geometry one day becomes the foundation of Math again. Anyone can participate in Geometry just with a stick or VR headset.
- Koshkin 2y ago
  I think everyone already knows some math before they start reading books on it.
senderista 2y ago
Ian Stewart's book is excellent for self-study and has some fascinating historical background. https://www.taylorfrancis.com/books/mono/10.1201/9781003213949/galois-theory-ian-stewart https://www.taylorfrancis.com/books/mono/10.1201/97810032139...
- susam 2y ago
  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 https://web.libera.chat/#bitwise [2] https://app.element.io/#/room/#bitwise:matrix.org https://app.element.io/#/room/#bitwise:matrix.org
  - jonathanyc 2y ago
    Thanks for this! I tried joining a few other channels on Libera like #robotics and ##typetheory but they’re kinda dead. Usenet Newsgroups are also all spam. Will check it out.
- nyankosensei 2y ago
  Another introduction from an historical point of view is “Galois Theory for Beginners: A Historical Perspective” by Jörg Bewersdorff https://bookstore.ams.org/view?ProductCode=STML/95 https://bookstore.ams.org/view?ProductCode=STML/95
- jaymzcampbell 2y ago
  I can agree with that, this is the primary textbook for the Open University's (excellent) Galois Theory MSc course (https://www.open.ac.uk/postgraduate/modules/m838 https://www.open.ac.uk/postgraduate/modules/m838. I really enjoyed my time on that course. I was also very interested in reading about the the original papers prior to the various advancements in mathematical thinking and notation that tend to reframe how the theory is taught today. For that I highly recommend Peter Neumanns "The mathematical writings of Evariste Galois" (https://ems.press/books/hem/102 https://ems.press/books/hem/102) – it has the french side by side with a direct English translation along with notes explaining the context and possible thought process (it also served as a fun way to read some more French whilst I was trying to learn the language).
jorgenveisdal 2y ago
Love this!
fredgrott 2y ago
Do not forget the numbers book covering history of numbers that Albert Einstein recommended....author is Tobias Dantzig...
- amai 2y ago
  I guess you mean this one: https://en.wikipedia.org/wiki/Number:_The_Language_of_Science https://en.wikipedia.org/wiki/Number:_The_Language_of_Scienc...
enugu 2y ago
There is a nice topological proof which gives a more direct and visual understanding what solving by radicals means. It is quite short but might take some time to absorb the concepts. https://jfeldbrugge.github.io/Galois-Theory/ https://jfeldbrugge.github.io/Galois-Theory/
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- deleted 2y ago
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artemonster 2y ago
I wonder, can you do an LLM in GF(2)?
- klyrs 2y ago
  Somebody did a paper on GF(3)... https://arxiv.org/abs/2402.17764 https://arxiv.org/abs/2402.17764
zengid 2y ago
ELI5 what Galois theory is?
- klyrs 2y ago
  My kid's 8 and still doesn't know what polynomials are. This one's gonna be tough. lol @ the coward who downvoted me without chiming in with a 5yo-digestible treatise on Galois theory
- zengid 2y ago
  how hard is it to say "an advanced theory about algebra", instead of downvoting?
- bubblyworld 2y ago
  Sometimes when you move a shape around in front of you, you end up with the same shape. Maths people call this symmetry, and have lots of names for different ways you can get back to the same shape. For instance, if you flip a square around you get back the same square. This is called reflective symmetry. If you spin a triangle around you sometimes end up with the same triangle. This one is rotational symmetry. Galois spent a lot of time thinking about numbers instead of shapes. What he realised is that when you add and multiply numbers in lots of different ways, you sometimes end up with the same number at the end. And sometimes different numbers, when added and multiplied in the same way, also give you the same number at the end. For instance, if you take 1, multiply it by itself and subtract 1, you get 1x1-1=0. If you do the same with -1, you get (-1)x(-1)-1=0. A different number, using the same pattern, gives us the same result. What we're seeing here is that there are some symmetries in numbers, not just shapes! Galois theory is all about the nitty gritty of how these number symmetries work, how to find them, and how to use them to do interesting mathematics.
  - zengid 2y ago
    this is a beautiful explanation! Thank you!
daitangio 2y ago
Btw, the life of Galois is quite interesting: he died very young, and was a quite clever mathematician…
klyrs 2y ago
> ... and I hope you can list all of the groups of order < 8 without having to think too hard. 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.
- gowld 2y ago
  Naming the groups of order 8 is harder than naming all the smaller groups.
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marshallward 2y ago
> I hope you can list all of the groups of order < 8 without having to think too hard. Welp, guess I'm out.
- Koshkin 2y ago
  https://en.wikipedia.org/wiki/List_of_small_groups https://en.wikipedia.org/wiki/List_of_small_groups
Koshkin 2y ago
Galois Theory For Beginners by John Stillwell is the shortest introduction that I've ever seen. https://www.scribd.com/document/81010821/GaloisTheoryForBeginners https://www.scribd.com/document/81010821/GaloisTheoryForBegi...
dmd 2y ago
https://chalkdustmagazine.com/blog/review-of-galois-knot-theory/ https://chalkdustmagazine.com/blog/review-of-galois-knot-the... is probably the best review.
- gowld 2y ago
  This is a joke not related to the submission or its topic.
broabprobe 2y ago
Danny O’Brien’s blog post A Touch of the Galois is my favorite writing on Galois, > 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.oblomovka.com/wp/2012/09/11/touch-of-the-galois/
- bhasi 2y ago
  The link appears to be down. Would love to read it though.
Venkatesh10 2y ago
The website is just plethora of knowledge and content in 90s design. Just pure bliss and I love it.
andyayers 2y ago
There are a few interesting places where Galois Theory touches on compilation/programming. 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).
- Koshkin 2y ago
  you must be writing lisp
  - andyayers 2y ago
    Not these days, but yes, years ago.
- SkiFire13 2y ago
  To extend on this, while abstract interpretation may sound a bit "abstract" (pun not intended), it is the basis for many techniques for software verification and compiler optimizations. At its core it basically allows you to soundly approximate the set of reachable states in a program, which in turn can be used to check that no "bad" state can be reached (for software verification) or that e.g. some checks are useless and can be removed (for compiler optimizations). Other applications include the new borrow checker for the Rust programming language, which is built on various dataflow passes to determine at each program point which variable is "live" and/or "borrowed".
dboreham 2y ago
Although a simple EE, I learned Galois Theory in college (coincidentally also in Edinburgh, although <other-university>). In 4th/final/senior year there were various elective classes including Advanced Mathematics which I chose as a kind of masochistic challenge. The class was very small, and it turned out taught by a "real mathematician" who commuted from the Mathematics department every day. Even though I've had a great deal of mathematics education I think this was the only time the teacher was someone who did mathematics all-in (as in he created new mathematics, published papers etc.) as opposed to someone who had the job of teaching some field (sic) in mathematics. 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".
- JadeNB 2y ago
  > 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. Presumably it helped to know that there was something beyond these folks' knowledge, but did it help in any more direct way?
  - dboreham 2y ago
    The main benefit was to do with how I conceived mathematics as a subject: In the before times, it appeared to be an infinite linear journey with subjects already studied in the set of "understood" things, and everything else in the "hard" set of things. Proceeding on the journey, subjects are expected to become harder and harder until eventually you're defeated and have to stop trying to learn more mathematics. After that class, my conception of mathematics was of an extensively cross-linked tree of subjects where seemingly unrelated fields connect to each other, and where although the number of fields is large, it is not infinite, and with effort and time everything can be understood if required. To be fair, that was the stated objective of the guy teaching the class. On day 1, he said "my objective is to teach you enough that you can understand whatever mathematics you need to understand in the future" and darn it he pretty much succeeded. In summary, more of an attitude than specific knowledge of anything in particular.
vladde 2y ago
That's an interesting way of holding a pen, never seen that before at 4:26 in https://ed-ac-uk.zoom.us/rec/play/qc1PCp8gTozfuRpMYKcTkPZQ2COysHZihM6jrWyQtYv_qUjirRDrRD9OLstYUcvQHHBBBy_vX5Dqk30u.o4mpBvrS8ZcNrt9k?canPlayFromShare=true&from=share_recording_detail&startTime=1610316133000&componentName=rec-play&originRequestUrl=https://ed-ac-uk.zoom.us/rec/share/HeDJyI0Eka0m8DLpxvPsHgCKh0k5QMQCpX3tRdbfn0mpjEk7IP88cKKix1WlsEdN.RrET5zAyw7zOw2V_?startTime%3D1610316133000 https://ed-ac-uk.zoom.us/rec/play/qc1PCp8gTozfuRpMYKcTkPZQ2C...
838592849 2y ago
Tom Leinster was the supervisor for my final year project a few years ago, he is a genius. He uses emacs!
jmount 2y ago
What do people think about the Edwards Galois Theory book?
will-burner 2y ago
Galois theory is the explanation and apex of theoretical math that you can motivate and talk about at a dinner table with people that don't even like math, lol. 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?
- gjm11 2y ago
  This isn't quite right -- Abel proved that there's no quintic formula before Galois came along. Galois theory gives a whole lot more insight, lets you understand why some quintics do have solutions in radicals, etc., but Galois doesn't (or at least shouldn't) get credited for proving that there isn't a quintic formula, because he wasn't the first to do that.
  - will-burner 2y ago
    Don't let the truth get in the way of a good story! hahahaha But yeah you're right edit: i don't recall Abel's proof, but Galois reformulation of what it means to be solvable by radicals, introducing the permutation group of the roots is the big thing in my mind.
    - arjvik 2y ago
      For a layman (I stopped short of Galois theory so far), what’s different about the permutation groups of quintic roots and above that leads to this?
javier_e06 2y ago
I was put through the ringer on Louis Leithold "Calculus, with analytic, geometry". Heavy heavy book. "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.

Galois Theory

467 points by mathgenius 2y ago

082349872349872 2y ago

In particular, abuse of Galois Theory makes it possible to reconcile Spinoza with Aquinas.

esafak 2y ago
Go on.
- koolala 2y ago
  Reconcile Liebniz too. I agree this sounds awesome.
  - 082349872349872 2y ago
    Leibniz does get a mention, but it'll already take a while to explain these two, so I'll leave his full reconciliation as an exercise for the reader ;)
    - koolala 2y ago
      ;) A mention is enough. Monads are LIFE!
      - 082349872349872 2y ago
        Every mathematical monad comes from an adjunction, which unfortunately implies they're not as windowless as Leibniz would like? (or does it merely imply that in the Leibnizian setting the structure is always external and never internal?) Can we make a mathematical monad out of the adjunction presented above? Let g(C) be the minimum G and c(G) be the maximum C in the model above, then we certainly have g(C) = g(c(g(C))) and c(G) = c(g(c(G))), which formally suggest we may be able to do something. Exercise: does the triple (T, μ, η) exist? what about (G, δ, ϵ) in the other direction? If they do exist, what are they, for Aquinas, Leibniz, and Spinoza?
novosel 2y ago
Why would they need to be reconciled? Or, indeed, Galoas abused?
- 082349872349872 2y ago
  They don't need to be reconciled, but they do differ, which suggests we might be able to find a framework which reconciles them. Galois Spinoza Aquinas in Google yields (for me at least) the following HN thread: https://news.ycombinator.com/item?id=39885475 https://news.ycombinator.com/item?id=39885475 , in which an attempt at "Algebraic Theology" not only provides such a subsuming (thanks to Galois Theory) framework, answering that particular question in the affirmative, but also raises many other questions which might be amusing to pursue. (I could summarise those Q's in this thread, if any of you all are more interested than the downvoters were) [I will summarise that thread further up in this one, but as it was months ago it may take me an hour or two to page everything back in]
082349872349872 2y ago
To expand[0] upon that claim: Aquinas, Leibniz, and Spinoza all agree that there is a God (G) that created[1] a Creation (C) which we are a part of. One major way Aquinas (and Leibniz) differ from Spinoza is in how determinate C may be. Spinoza says C is determined by G; Aquinas says there are many possible C's for any given G. (Leibniz splits the difference and says there are many possible C's, but in our particular case, our G has created the best[2] possible C.) The reconciliation: let G and C be in an adjoint relationship, such that we have functions picking out the maximum C any given G may create, and the minimum G that can create any given C. Now, if you are Aquinas, G is omnipotent, and hence has the possibility to create other C's, but our C is the maximum[3] one. On the other hand, if you are Spinoza, G determines C[4], so it is trivially maximal. (The maximum of a singleton being the unique element) Does that make sense? Question: do there exist Gods that are incapable of creating any creation, or Creations that are impossible for any god to create? If so, need we replace "maximum" and "minimum" above by LUB and GLB? What other situations (eg. gods or creations being only domains rather than lattices) would also require further abstraction? :: :: :: [0] my apologies for any non-standard notation. I tried to find a survey paper on "Algebraic Theology" so I could follow the existing notation, but failed to find any concrete instances in this (currently only platonic?) field. [1] Spinoza is accused of "pantheism": the heresy of identifying God and the Creation. Reading him according to a Galois-theoretic model, he would be innocent of this heresy, for when he says "God, or, the Universe", he is simply using metonymy, for in his model God and Creation are dual, so (being in a 1:1 relationship) one determines the other. Note that in general, not only are duals not identical, they're not even isomorphic. [2] but cf Voltaire, Candide (1759) [3] if you are Leibniz, the order in which it is maximal corresponds to the traditional "worst" "better" "best" order. I don't know Leibniz well enough to say if he had a total, or only partial, order in mind; presumably in his model if there are several maximal "best" creations they would all be isomorphic? Exercise for the reader: work out the Leibnizian metaphysical adjunction. [4] turning this arrow around, C determines G, which explains why Einstein would say he believed in the "God of Spinoza", and chose to base his research by thinking about C, unlike the medieval colleagues of Aquinas, who spent a lot of time and effort trying to work the arrow in the other direction, hoping to come to conclusions about C in starting by thinking about G.
- 2y ago

rpmw 2y ago

I will always remember Galois theory as the punchline to my Abstract Algebra courses in college. Galois was a brilliant math mind, and I'm curious what else he would have contributed had he not died at 20 in a duel. https://en.wikipedia.org/wiki/%C3%89variste_Galois https://en.wikipedia.org/wiki/%C3%89variste_Galois

mcfunley 2y ago
Great to see this course material public. It's a real missed opportunity though to not mention that Galois wrote a lot of it down staying up all night before being shot.
- bubble12345 2y ago
  That's a common myth. See this paper referenced in the wikipedia article: Rothman, Tony (1982). "Genius and Biographers: The Fictionalization of Evariste Galois". The American Mathematical Monthly. 89 (2): 84–106. doi:10.2307/2320923. JSTOR 2320923
  - ForOldHack 2y ago
    Quite the clickbait. You can only access it from the pay site, or unless you can get a school library to access it, which I will do. Only the first page is available free.
    - bubble12345 2y ago
      Or you can paste the following JSTOR link into sci-hub. https://www.jstor.org/stable/2320923 https://www.jstor.org/stable/2320923
agumonkey 2y ago
I'd be genuinely curious to have a chat with similar minded people. They must see the world quite differently to be able to fork a new path in hard maths mostly on their own ..

frakt0x90 2y ago

My second semester of algebra had a section on Galois theory and I remember thinking it was abstract nonsense and I didn't get it. I'm actually interested in going through this to see if my perspective has changed.

openasocket 2y ago
It's all abstract nonsense until it starts having practical results lol :) The main motivation for Galois theory was proving the insolvability of the quintic. For those not aware, there is a general formula solving the quadratic equation (i.e. solving ax^2 + bx + c = 0). That formula has been known for millenia. With effort, mathematicians found a formula for the cubic (i.e. solving ax^3 + bx^2 + cx + d = 0), and even the quartic (order 4 polynomials). But no one was able to come up with a closed-form solution to the quintic. Galois and Abel eventually proved that a quntic formula DOES NOT EXIST. At least, it cannot be expressed in terms of addition, subtraction, multiplication, division, exponents and roots. You can even identify specific equations that have roots that cannot be expressed in those forms, for example x^5 - x + 1. I took an entire course on it that went through the proof. It's actually very interesting. It sets up this deep correspondence between groups and fields, to the point that any theory about groups can be translated into a theory about fields and vice versa. And it provides this extremely powerful set of tools for analyzing symmetries. The actual proof is actually really anticlimactic. You have all these deep proofs about the structures of roots of polynomial equations, and at the very end you just see that the structures of symmetries of certain polynomials of degree 5 (like x^5 - x + 1) don't follow the same symmetries that the elementary mathematical operations have. Literally, the field of solutions to that polynomial doesn't map to a solvable group: https://en.wikipedia.org/wiki/Solvable_group https://en.wikipedia.org/wiki/Solvable_group In almost the same breadth, it also proves that it is impossible to trisect an angle using just a compass and straightedge, a problem that had been puzzling mathematicians for millennia. It's actually almost disappointing: we spent then entire course just defining groups and fields and field extensions and all this other "abstract nonsense". And once all of those definitions are out of the way, the proof of the insolvability of the quintic takes 10 minutes, same for the proof of impossibility of trisecting an angle.
- 2y ago

VyseofArcadia 2y ago

Chapter 1 is brilliant. 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.

gowld 2y ago
Hot take: "18th Century" mathematics was intiuitive and informal, to the point that it was inconsistent. The 19th and 20th Centuries added rigor and formalism (and elitism) and devalued intuition, to the point that it begame uninterpretable to most. The 21st Century's major contribution to mathematics (including YouTube! and conversational style writing) was to bring back intuition, with the backing of formal foundations.
- Agingcoder 2y ago
  I like your idea. I was schooled in abstract 20th century math - indeed YouTube is the opposite, and it’s a good thing. One of my math teachers was once talking to Jean Dieudonné https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9 https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9 who was part of the Bourbaki group and asked him why on earth he insisted on inflicting raw dry theory to the world with no intuition , when his day job involved drawing ideas all day long ! Edit: interestingly enough, one of my colleagues thinks very strongly that intuition should not be shared, and the path to intuition should be walked by everyone so that they ´ Make their own mental images ´. I guess that there’s a tradeoff between making things accessible, and deeply understood, but I don’t know what to make of his opinion.

gowld 2y ago

Notes, Videos, and Problems: https://www.maths.ed.ac.uk/~tl/galois/#notes https://www.maths.ed.ac.uk/~tl/galois/#notes Direct link to PDF of notes: https://arxiv.org/pdf/2408.07499 https://arxiv.org/pdf/2408.07499

mkw5053 2y ago

A few years ago, I led a study group through A Book of Abstract Algebra by Charles C Pinter. It culminated in Galois Theory and was one of the best books I've ever used in a math study group.

mkw5053 2y ago
Also, it assumes little to no advanced math knowledge and can be found for free online :)
sifar 2y ago
I second this book, really accessible. It taught me Abstract Algebra when I was learning it by myself.
bosquefrio 2y ago
I have fond memories of reading this book in college. I enjoyed it immensely. I also remember reading the section at the back of the book about Galois. There was also an entertaining section about the history of solving the roots of polynomial equations and in particular solving equations of arbitrary order.

PreInternet01 2y ago

[removed by author]

ogogmad 2y ago
I don't think this is Galois theory. Galois theory is about "The Fundamental Theorem of Galois Theory", which states that there is a nice mapping from field-extensions to group-extensions, where the resulting groups are usually finite. When the resulting groups are finite (as they usually are), many problems involving field extensions can be solved using brute-force search. Galois fields happen to be something else named in honour of Galois.
- koolala 2y ago
  Galois Fields yeah. Being able to store Rational Numbers in a computer. "Originally, the theory had been developed for algebraic equations whose coefficients are rational numbers."
  - matt-noonan 2y ago
    That is definitely not what Galois Fields are about.
    - koolala 2y ago
      https://en.m.wikipedia.org/wiki/Galois_theory#Permutation_group_approach https://en.m.wikipedia.org/wiki/Galois_theory#Permutation_gr... That quote about equations with rational numbers was from here. Galois didn't have a computer of course. Rational Numbers and trisecting an angle sound related.
      - wging 2y ago
        The quote is followed immediately by this: "It extends naturally to equations with coefficients *in any field*, but this will not be considered in the simple examples below." Emphasis on 'in any field' is mine. Among the other fields that can be considered include the Galois fields, which are another name for finite fields. (There are also infinite fields other than the rationals, so 'in any field' does not just mean Galois fields/finite fields.) https://en.wikipedia.org/wiki/Finite_field https://en.wikipedia.org/wiki/Finite_field Galois fields have nothing to do with being able to represent rational numbers in a computer: elements of a finite field aren't even rational numbers.

ogogmad 2y ago

Interestingly, there's a close connection between the "Fundamental Theorem of Galois Theory" and the "Fundamental Theorem of Covering Spaces".

HPsquared 2y ago

For non-math people, is this "simple Wikipedia" article about right? I've always seen Galois theory listed in mathematics courses and wondered what it is, speaking as a humble engineer. https://simple.m.wikipedia.org/wiki/Galois_theory https://simple.m.wikipedia.org/wiki/Galois_theory

gnulinux 2y ago
Yes and no, it's a bit too simplistic and doesn't explain the actual "why" of Galois Theory, just the how. The brilliant insight Galois figured out is that there is a fundamental connection between fields and groups, but that is just the "technique" with which he solved the problem. The "why even bother" is a bit more complex but simply put Galois wanted to establish a criterion to determine what polynomials are solvable or unsolvable in which fields. E.g. we know x^2=-1 is solvable in C with x=i but not in real numbers. Can we generalize that proof to such a degree that we can mechanistically run it for arbitrary polynomials in arbitrary fields?
fredilo 2y ago
What it states is correct and it gives you a good overview over what you do in a Galois theory course. It does, however, not give you an idea of why this is interesting. When just reading that article one might get the idea that some mathematicians just had too much free time. I tried to motivate the questions leading to Galois Theory in https://news.ycombinator.com/item?id=41258726 https://news.ycombinator.com/item?id=41258726 in a way that is hopefully accessible to more down-to-earth programmers and engineers.
- fredilo 2y ago
  I should probably add why I think the motivation is so important here. For pure engineers, numbers are a tool. They ask: What can I build with numbers? Pure mathematicians ask a different question. They are interested in the limits of numbers. They ask: What can I not build with numbers? Studying these two questions is deeply related but also a constant source of frustration for engineers taking math courses designed for mathematicians by mathematicians. Galois theory, is a theory of "no". It ultimately serves to answer several "Can I build this?" questions with no. This makes it very interesting to pure mathematicians. However, for pure engineers that are looking for numeric machine parts that can be assembled in other useful ways to actually build something... Galois theory can be quite disappointing.

raldi 2y ago

Clicked around for a few minutes and couldn’t find a sentence beginning, “Galois Theory is…”

chii 2y ago
https://www.maths.ed.ac.uk/~tl/galois/ https://www.maths.ed.ac.uk/~tl/galois/ it's a link to the course, and there's an introduction right at the beginning of the course. Unfortunately, it's a video, rather than text, but there's at least transcript. Even wikipedia acknowledges that it is not a simple subject: https://simple.wikipedia.org/wiki/Galois_theory#Disclaimer https://simple.wikipedia.org/wiki/Galois_theory#Disclaimer ;so hoping you can get an understanding of it in a few sentences is probably asking too much tbh.
skulk 2y ago
Someone could tell you that Galois theory is the study of field extensions and the structure of their automorphism groups but is that really going to help?
- gowld 2y ago
  The fundamental theorem of Galois theory reduces certain problems in field theory (like finding roots of polynomials) to group theory (counting permutations and symmetries), which makes them simpler and easier to understand and solve (or prove non-solvable). Galois theory is the proof and applications of this theorem, and related topics. Key to this theory is being more methodical about extending the rational numbers into the real numbers, by introducing new numbers one at a time, instead of all at once. One of the immediate discoveries in beginning this study is the fact that in many common cases you cannot add just one numbers one at a time, but must add 2 or more numbers at once. These sets of numbers are called conjugates, and have the interesting property that even though you can prove how many must exist and that they are distinct from each other, they are otherwise identical except in the arbitray names you give them.
taneliv 2y ago
I watched one of the videos, titled "Galois groups, intuitively".[1] It is about 18 minutes long and gave me a somewhat understandable overview what Galois groups are (not the theory yet!). I suppose if you have the time to spare, at least it should give you an idea if the topic is of interest and whether you need to remind yourself of some mathematical concepts to be able to study rest of the material, or if it is too elementary for you and a shorter treatise from elsewhere would be preferable. (I actually thought I had learned something about Galois theory in University algebra, but either I hadn't, or I've forgotten more than I wanted to admit. Which is to say, if you watch the video, your mileage may vary!) [1] https://ed-ac-uk.zoom.us/rec/share/_I-EeZA8_399ArdZ1GyKtM_rDWHI8uE_n4YysAZwJA1EF3Ix2cgRkxoABrZEU0td.sSYj18b7bczdplw0?startTime=1610319766000 https://ed-ac-uk.zoom.us/rec/share/_I-EeZA8_399ArdZ1GyKtM_rD... (link copied from the page, I hope it works)

Joker_vD 2y ago

> But then you realize something genuinely weird: There’s nothing you can do to distinguish i from −i. 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?

solveit 2y ago
Stack the vectors up so it's a matrix and take the determinant. The sign tells you which one it is.
- gowld 2y ago
  No, it does not. The determinant tells you whether two bases have the same or different handedness, not which one is "left" or "right".
  - lanstin 2y ago
    It's 2 cosets, one is arbitrarily left handed and the other arbitrarily right handed. If you are in an orientable space :) if not, then there's no global concept of left or right. a
    - fredilo 2y ago
      Formulated differently, you cannot determine left- and right-handedness but you can determine same-handedness.
gowld 2y ago
No, it's impossible, even in principle, to answer that question. You can draw a picture for either answer. "Left" and "right" are a dipole. Neither one can exist without the other, and they are symmetric. It's the same issue as we have with the conjugates discussed in Galois Theory. In fact, in an algebraic (non-ordered/arithmetic/analytic) perspective, it's misleading to use the symbols + and - to label the conjugates in field extensions like sqrt2 and i. Left and Right are better names than + and - for those conjugate pairs. Only when we impose an arithmetic ordering (which is not needed in the theory of algebraic equalities) is it meaningful to use + and -: -sqrt(x) < 0 < +sqrt(x), where x is a positive real number. (and when x is a negative real number, we immediately see the problem with - again: -i and i are not separable via ordering with respect to 0.)

revskill 2y ago

The problem with many mathematics books, is it uses Math to teach Math !!! 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 :)

koolala 2y ago
Painting, Sculpture, Music, Art is a Language. We need the Toki Pona of Math. I hope Geometry one day becomes the foundation of Math again. Anyone can participate in Geometry just with a stick or VR headset.
Koshkin 2y ago
I think everyone already knows some math before they start reading books on it.

senderista 2y ago

Ian Stewart's book is excellent for self-study and has some fascinating historical background. https://www.taylorfrancis.com/books/mono/10.1201/9781003213949/galois-theory-ian-stewart https://www.taylorfrancis.com/books/mono/10.1201/97810032139...

susam 2y ago
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 https://web.libera.chat/#bitwise [2] https://app.element.io/#/room/#bitwise:matrix.org https://app.element.io/#/room/#bitwise:matrix.org
- jonathanyc 2y ago
  Thanks for this! I tried joining a few other channels on Libera like #robotics and ##typetheory but they’re kinda dead. Usenet Newsgroups are also all spam. Will check it out.
nyankosensei 2y ago
Another introduction from an historical point of view is “Galois Theory for Beginners: A Historical Perspective” by Jörg Bewersdorff https://bookstore.ams.org/view?ProductCode=STML/95 https://bookstore.ams.org/view?ProductCode=STML/95
jaymzcampbell 2y ago
I can agree with that, this is the primary textbook for the Open University's (excellent) Galois Theory MSc course (https://www.open.ac.uk/postgraduate/modules/m838 https://www.open.ac.uk/postgraduate/modules/m838. I really enjoyed my time on that course. I was also very interested in reading about the the original papers prior to the various advancements in mathematical thinking and notation that tend to reframe how the theory is taught today. For that I highly recommend Peter Neumanns "The mathematical writings of Evariste Galois" (https://ems.press/books/hem/102 https://ems.press/books/hem/102) – it has the french side by side with a direct English translation along with notes explaining the context and possible thought process (it also served as a fun way to read some more French whilst I was trying to learn the language).

jorgenveisdal 2y ago

Love this!

fredgrott 2y ago

Do not forget the numbers book covering history of numbers that Albert Einstein recommended....author is Tobias Dantzig...

amai 2y ago
I guess you mean this one: https://en.wikipedia.org/wiki/Number:_The_Language_of_Science https://en.wikipedia.org/wiki/Number:_The_Language_of_Scienc...

enugu 2y ago

There is a nice topological proof which gives a more direct and visual understanding what solving by radicals means. It is quite short but might take some time to absorb the concepts. https://jfeldbrugge.github.io/Galois-Theory/ https://jfeldbrugge.github.io/Galois-Theory/

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artemonster 2y ago

I wonder, can you do an LLM in GF(2)?

klyrs 2y ago
Somebody did a paper on GF(3)... https://arxiv.org/abs/2402.17764 https://arxiv.org/abs/2402.17764

zengid 2y ago

ELI5 what Galois theory is?

klyrs 2y ago
My kid's 8 and still doesn't know what polynomials are. This one's gonna be tough. lol @ the coward who downvoted me without chiming in with a 5yo-digestible treatise on Galois theory
zengid 2y ago
how hard is it to say "an advanced theory about algebra", instead of downvoting?
bubblyworld 2y ago
Sometimes when you move a shape around in front of you, you end up with the same shape. Maths people call this symmetry, and have lots of names for different ways you can get back to the same shape. For instance, if you flip a square around you get back the same square. This is called reflective symmetry. If you spin a triangle around you sometimes end up with the same triangle. This one is rotational symmetry. Galois spent a lot of time thinking about numbers instead of shapes. What he realised is that when you add and multiply numbers in lots of different ways, you sometimes end up with the same number at the end. And sometimes different numbers, when added and multiplied in the same way, also give you the same number at the end. For instance, if you take 1, multiply it by itself and subtract 1, you get 1x1-1=0. If you do the same with -1, you get (-1)x(-1)-1=0. A different number, using the same pattern, gives us the same result. What we're seeing here is that there are some symmetries in numbers, not just shapes! Galois theory is all about the nitty gritty of how these number symmetries work, how to find them, and how to use them to do interesting mathematics.
- zengid 2y ago
  this is a beautiful explanation! Thank you!

daitangio 2y ago

Btw, the life of Galois is quite interesting: he died very young, and was a quite clever mathematician…

klyrs 2y ago

> ... and I hope you can list all of the groups of order < 8 without having to think too hard. 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.

gowld 2y ago
Naming the groups of order 8 is harder than naming all the smaller groups.

deleted 2y ago

[deleted]

marshallward 2y ago

> I hope you can list all of the groups of order < 8 without having to think too hard. Welp, guess I'm out.

Koshkin 2y ago
https://en.wikipedia.org/wiki/List_of_small_groups https://en.wikipedia.org/wiki/List_of_small_groups

Koshkin 2y ago

Galois Theory For Beginners by John Stillwell is the shortest introduction that I've ever seen. https://www.scribd.com/document/81010821/GaloisTheoryForBeginners https://www.scribd.com/document/81010821/GaloisTheoryForBegi...

dmd 2y ago

https://chalkdustmagazine.com/blog/review-of-galois-knot-theory/ https://chalkdustmagazine.com/blog/review-of-galois-knot-the... is probably the best review.

gowld 2y ago
This is a joke not related to the submission or its topic.

broabprobe 2y ago

Danny O’Brien’s blog post A Touch of the Galois is my favorite writing on Galois, > 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.oblomovka.com/wp/2012/09/11/touch-of-the-galois/

bhasi 2y ago
The link appears to be down. Would love to read it though.

Venkatesh10 2y ago

The website is just plethora of knowledge and content in 90s design. Just pure bliss and I love it.

andyayers 2y ago

There are a few interesting places where Galois Theory touches on compilation/programming. 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).

Koshkin 2y ago
you must be writing lisp
- andyayers 2y ago
  Not these days, but yes, years ago.
SkiFire13 2y ago
To extend on this, while abstract interpretation may sound a bit "abstract" (pun not intended), it is the basis for many techniques for software verification and compiler optimizations. At its core it basically allows you to soundly approximate the set of reachable states in a program, which in turn can be used to check that no "bad" state can be reached (for software verification) or that e.g. some checks are useless and can be removed (for compiler optimizations). Other applications include the new borrow checker for the Rust programming language, which is built on various dataflow passes to determine at each program point which variable is "live" and/or "borrowed".

dboreham 2y ago

Although a simple EE, I learned Galois Theory in college (coincidentally also in Edinburgh, although <other-university>). In 4th/final/senior year there were various elective classes including Advanced Mathematics which I chose as a kind of masochistic challenge. The class was very small, and it turned out taught by a "real mathematician" who commuted from the Mathematics department every day. Even though I've had a great deal of mathematics education I think this was the only time the teacher was someone who did mathematics all-in (as in he created new mathematics, published papers etc.) as opposed to someone who had the job of teaching some field (sic) in mathematics. 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".

JadeNB 2y ago
> 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. Presumably it helped to know that there was something beyond these folks' knowledge, but did it help in any more direct way?
- dboreham 2y ago
  The main benefit was to do with how I conceived mathematics as a subject: In the before times, it appeared to be an infinite linear journey with subjects already studied in the set of "understood" things, and everything else in the "hard" set of things. Proceeding on the journey, subjects are expected to become harder and harder until eventually you're defeated and have to stop trying to learn more mathematics. After that class, my conception of mathematics was of an extensively cross-linked tree of subjects where seemingly unrelated fields connect to each other, and where although the number of fields is large, it is not infinite, and with effort and time everything can be understood if required. To be fair, that was the stated objective of the guy teaching the class. On day 1, he said "my objective is to teach you enough that you can understand whatever mathematics you need to understand in the future" and darn it he pretty much succeeded. In summary, more of an attitude than specific knowledge of anything in particular.

vladde 2y ago

That's an interesting way of holding a pen, never seen that before at 4:26 in https://ed-ac-uk.zoom.us/rec/play/qc1PCp8gTozfuRpMYKcTkPZQ2COysHZihM6jrWyQtYv_qUjirRDrRD9OLstYUcvQHHBBBy_vX5Dqk30u.o4mpBvrS8ZcNrt9k?canPlayFromShare=true&from=share_recording_detail&startTime=1610316133000&componentName=rec-play&originRequestUrl=https://ed-ac-uk.zoom.us/rec/share/HeDJyI0Eka0m8DLpxvPsHgCKh0k5QMQCpX3tRdbfn0mpjEk7IP88cKKix1WlsEdN.RrET5zAyw7zOw2V_?startTime%3D1610316133000 https://ed-ac-uk.zoom.us/rec/play/qc1PCp8gTozfuRpMYKcTkPZQ2C...

838592849 2y ago

Tom Leinster was the supervisor for my final year project a few years ago, he is a genius. He uses emacs!

jmount 2y ago

What do people think about the Edwards Galois Theory book?

will-burner 2y ago

Galois theory is the explanation and apex of theoretical math that you can motivate and talk about at a dinner table with people that don't even like math, lol. 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?

gjm11 2y ago
This isn't quite right -- Abel proved that there's no quintic formula before Galois came along. Galois theory gives a whole lot more insight, lets you understand why some quintics do have solutions in radicals, etc., but Galois doesn't (or at least shouldn't) get credited for proving that there isn't a quintic formula, because he wasn't the first to do that.
- will-burner 2y ago
  Don't let the truth get in the way of a good story! hahahaha But yeah you're right edit: i don't recall Abel's proof, but Galois reformulation of what it means to be solvable by radicals, introducing the permutation group of the roots is the big thing in my mind.
  - arjvik 2y ago
    For a layman (I stopped short of Galois theory so far), what’s different about the permutation groups of quintic roots and above that leads to this?

javier_e06 2y ago

I was put through the ringer on Louis Leithold "Calculus, with analytic, geometry". Heavy heavy book. "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.