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Galois Theory

467 点作者 mathgenius9 个月前

32 条评论

dboreham9 个月前
Although a simple EE, I learned Galois Theory in college (coincidentally also in Edinburgh, although &lt;other-university&gt;). In 4th&#x2F;final&#x2F;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 &quot;real mathematician&quot; who commuted from the Mathematics department every day. Even though I&#x27;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.<p>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&#x27;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 &quot;I never used a single thing I learned in college&quot;.
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senderista9 个月前
Ian Stewart&#x27;s book is excellent for self-study and has some fascinating historical background.<p><a href="https:&#x2F;&#x2F;www.taylorfrancis.com&#x2F;books&#x2F;mono&#x2F;10.1201&#x2F;9781003213949&#x2F;galois-theory-ian-stewart" rel="nofollow">https:&#x2F;&#x2F;www.taylorfrancis.com&#x2F;books&#x2F;mono&#x2F;10.1201&#x2F;97810032139...</a>
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VyseofArcadia9 个月前
Chapter 1 is brilliant.<p>I&#x27;ve been shouting from the rooftops for years that math[0] courses need more <i>context</i>. 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.<p>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.<p>Kudos to the author for chapter 1 (and probably the rest, but chapter 1 is all I&#x27;ve had time to skim).<p>[0] And honestly, nearly every subject.
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will-burner9 个月前
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&#x27;t even like math, lol.<p>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&#x27;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)?<p>It&#x27;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.<p>Anyways, the punchline is that there&#x27;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.<p>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?
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rpmw9 个月前
I will always remember Galois theory as the punchline to my Abstract Algebra courses in college. Galois was a brilliant math mind, and I&#x27;m curious what else he would have contributed had he not died at 20 in a duel.<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;%C3%89variste_Galois" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;%C3%89variste_Galois</a>
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mkw50539 个月前
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&#x27;ve ever used in a math study group.
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HPsquared9 个月前
For non-math people, is this &quot;simple Wikipedia&quot; article about right? I&#x27;ve always seen Galois theory listed in mathematics courses and wondered what it is, speaking as a humble engineer. <a href="https:&#x2F;&#x2F;simple.m.wikipedia.org&#x2F;wiki&#x2F;Galois_theory" rel="nofollow">https:&#x2F;&#x2F;simple.m.wikipedia.org&#x2F;wiki&#x2F;Galois_theory</a>
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frakt0x909 个月前
My second semester of algebra had a section on Galois theory and I remember thinking it was abstract nonsense and I didn&#x27;t get it. I&#x27;m actually interested in going through this to see if my perspective has changed.
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andyayers9 个月前
There are a few interesting places where Galois Theory touches on compilation&#x2F;programming.<p>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.<p>Also (perhaps a bit of stretch) it&#x27;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&#x27;s poison &amp; undef, or NaNs).
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Joker_vD9 个月前
&gt; But then you realize something genuinely weird: <i>There’s nothing you can do to distinguish</i> i <i>from −</i>i.<p>Relatedly, to this day I still don&#x27;t know how distinguish a left-handed coordinate system from the right-handed one <i>purely algebraically</i>. Is the basis [(1,0,0), (0,1,0), (0,0,1)] left- or right-handed? I don&#x27;t know without a picture! Does anyone?
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broabprobe9 个月前
Danny O’Brien’s blog post A Touch of the Galois is my favorite writing on Galois,<p>&gt; 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.<p><a href="https:&#x2F;&#x2F;www.oblomovka.com&#x2F;wp&#x2F;2012&#x2F;09&#x2F;11&#x2F;touch-of-the-galois&#x2F;" rel="nofollow">https:&#x2F;&#x2F;www.oblomovka.com&#x2F;wp&#x2F;2012&#x2F;09&#x2F;11&#x2F;touch-of-the-galois&#x2F;</a>
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Koshkin9 个月前
<i>Galois Theory For Beginners</i> by John Stillwell is the shortest introduction that I&#x27;ve ever seen.<p><a href="https:&#x2F;&#x2F;www.scribd.com&#x2F;document&#x2F;81010821&#x2F;GaloisTheoryForBeginners" rel="nofollow">https:&#x2F;&#x2F;www.scribd.com&#x2F;document&#x2F;81010821&#x2F;GaloisTheoryForBegi...</a>
enugu9 个月前
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.<p><a href="https:&#x2F;&#x2F;jfeldbrugge.github.io&#x2F;Galois-Theory&#x2F;" rel="nofollow">https:&#x2F;&#x2F;jfeldbrugge.github.io&#x2F;Galois-Theory&#x2F;</a>
vladde9 个月前
That&#x27;s an interesting way of holding a pen, never seen that before<p>at 4:26 in <a href="https:&#x2F;&#x2F;ed-ac-uk.zoom.us&#x2F;rec&#x2F;play&#x2F;qc1PCp8gTozfuRpMYKcTkPZQ2COysHZihM6jrWyQtYv_qUjirRDrRD9OLstYUcvQHHBBBy_vX5Dqk30u.o4mpBvrS8ZcNrt9k?canPlayFromShare=true&amp;from=share_recording_detail&amp;startTime=1610316133000&amp;componentName=rec-play&amp;originRequestUrl=https:&#x2F;&#x2F;ed-ac-uk.zoom.us&#x2F;rec&#x2F;share&#x2F;HeDJyI0Eka0m8DLpxvPsHgCKh0k5QMQCpX3tRdbfn0mpjEk7IP88cKKix1WlsEdN.RrET5zAyw7zOw2V_?startTime%3D1610316133000" rel="nofollow">https:&#x2F;&#x2F;ed-ac-uk.zoom.us&#x2F;rec&#x2F;play&#x2F;qc1PCp8gTozfuRpMYKcTkPZQ2C...</a>
javier_e069 个月前
I was put through the ringer on Louis Leithold &quot;Calculus, with analytic, geometry&quot;. Heavy heavy book.<p>&quot;Do the exercises&quot; teacher echoed over and over. I read the chapter, I followed the examples and proceed to the first problem in the unit.<p>My answer was 64<p>I go to the end of the book and the answer was 2 1&#x2F;4<p>I would try to reverse engineer the 2 and 1&#x2F;4 to original problem... Nothing!<p>I would ask a friend to the problem with me.. her answer was 16.<p>Maybe divide by 8? that gets us 2, we are closer? Right. Why divide by 8? I don&#x27;t know!<p>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.
gowld9 个月前
Notes, Videos, and Problems: <a href="https:&#x2F;&#x2F;www.maths.ed.ac.uk&#x2F;~tl&#x2F;galois&#x2F;#notes" rel="nofollow">https:&#x2F;&#x2F;www.maths.ed.ac.uk&#x2F;~tl&#x2F;galois&#x2F;#notes</a><p>Direct link to PDF of notes: <a href="https:&#x2F;&#x2F;arxiv.org&#x2F;pdf&#x2F;2408.07499" rel="nofollow">https:&#x2F;&#x2F;arxiv.org&#x2F;pdf&#x2F;2408.07499</a>
klyrs9 个月前
&gt; ... and I hope you can list all of the groups of order &lt; 8 without having to think too hard.<p>Early morning reaction: oh god I&#x27;ve forgotten all of my group theory, this is <i>bad</i>.<p>After lunch: oh, right, there&#x27;s only two composite numbers below 8.
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ogogmad9 个月前
Interestingly, there&#x27;s a close connection between the &quot;Fundamental Theorem of Galois Theory&quot; and the &quot;Fundamental Theorem of Covering Spaces&quot;.
dmd9 个月前
<a href="https:&#x2F;&#x2F;chalkdustmagazine.com&#x2F;blog&#x2F;review-of-galois-knot-theory&#x2F;" rel="nofollow">https:&#x2F;&#x2F;chalkdustmagazine.com&#x2F;blog&#x2F;review-of-galois-knot-the...</a> is probably the best review.
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fredgrott9 个月前
Do not forget the numbers book covering history of numbers that Albert Einstein recommended....author is Tobias Dantzig...
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daitangio9 个月前
Btw, the life of Galois is quite interesting: he died very young, and was a quite clever mathematician…
marshallward9 个月前
&gt; I hope you can list all of the groups of order &lt; 8 without having to think too hard.<p>Welp, guess I&#x27;m out.
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Venkatesh109 个月前
The website is just plethora of knowledge and content in 90s design. Just pure bliss and I love it.
8385928499 个月前
Tom Leinster was the supervisor for my final year project a few years ago, he is a genius.<p>He uses emacs!
raldi9 个月前
Clicked around for a few minutes and couldn’t find a sentence beginning, “Galois Theory is…”
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jmount9 个月前
What do people think about the Edwards Galois Theory book?
revskill9 个月前
The problem with many mathematics books, is it uses Math to teach Math !!!<p>OK, it&#x27;s fine in some cases, but it&#x27;s like a gatekeeping itself, because in order to understand Math, you need to understand Math :)
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artemonster9 个月前
I wonder, can you do an LLM in GF(2)?
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zengid9 个月前
ELI5 what Galois theory is?
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jorgenveisdal9 个月前
Love this!
PreInternet019 个月前
[removed by author]
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0823498723498729 个月前
In particular, abuse of Galois Theory makes it possible to reconcile Spinoza with Aquinas.
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