[HN Gopher] Galois Groups and the Symmetries of Polynomials
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Galois Groups and the Symmetries of Polynomials
Author : nsoonhui
Score : 84 points
Date : 2021-08-04 08:50 UTC (14 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| shreyshnaccount wrote:
| I'm gonna self study group theory, at first using the book
| 'visual group theory' and figuring out the rest later. smart
| people of HN, give advice, resources please. thanks in advance
| gmadsen wrote:
| I've done the same, I'm a roboticist by training, but I've
| successfully self studied abstract algebra, topology, and
| complex analysis.
|
| It certainly helped that I took proof based real analysis and
| linear algebra in school. Writing proofs is really hard to
| learn without a feedback loop of a professor. So if you don't
| have any experience with that, I'd start with an intro to
| proofs book first.
|
| but the group theory material is generally easier than
| something like topology or functional analysis, which I am
| certain I was only able to do from finding like minded people
| in the mathematics sub reddit and did basically a mock semester
| class on discord.
|
| specific to algebra, I really like the classic Pinter book, "A
| book of Abstract Algebra", which does a great job of mixing in
| exposition and history that was really beneficial for self
| study.
| abetusk wrote:
| My education on groups is by far and away deficient but one of
| the better books I came across is called "Fundamental
| Algorithms for Permutation Groups" by G. Butler [0]. It's kind
| of old (1991) but, in my opinion, excellent, especially for
| some one like me that's concerned with computation, run-time,
| algorithms, etc.
|
| There's also a Akos Seress book, "Permutation Group
| Algorithms", which fills in some mathematical details and
| proofs but, notably, doesn't have one line of pseudo-code in
| the book.
|
| Both focus on algorithms, and permutation groups in particular,
| which is just a small subset of group theory.
|
| [0] https://www.springer.com/gp/book/9783540549550
| sdenton4 wrote:
| For the permutation group in particular, it's hard to beat
| Sagan's 'The Symmetric Group.' It's a bit more recent, and
| covers lots of really beautiful topics, and really shows off
| the nice connections between the combinatorics of S_n and the
| representation theory.
|
| https://www.springer.com/gp/book/9780387950679
| abetusk wrote:
| Thanks for the recommendation.
|
| Just briefly reviewing it, this is the kind of book I tend
| to shy away from. I'm sure it's a good reference for a
| particular type of group theory but I doubt I would use
| this type of book for anything but a reference.
|
| I can't find any mention of "strong generating set" or
| algorithmic considerations of how to actually compute with
| permutation groups. As with many of the Springer books, it
| feels like a collection of scattered knowledge without a
| cohesive narrative.
|
| It's completely possible I'm missing something with this
| book, especially since I haven't read it, and that for more
| mathematically minded folks, this book is a better
| introduction.
| zant wrote:
| Hey this is really cool. I want to start doing the same thing.
|
| Visual group theory is a really nice book for intuition. Also,
| the YouTube series "Essence of Group Theory" can help in this
| same line.
|
| What I also want to do while self learning is formalizing some
| theorems and definitions in Lean. Even just looking at how
| they're already defined in mathlib [1] can be of great help
| when internalizing the concepts.
|
| https://github.com/leanprover-community/mathlib/blob/292e3fa...
| jimsimmons wrote:
| Side question: what do you think of Lean? I'm not sure what
| people who know this stuff think of it. I've read comments
| that say Lean is inferior to OCaml and it simply hyped by
| celebrity
| zomglings wrote:
| I really like Abstract Algebra by Dummit and Foote.
|
| Get an old, used, international edition.
|
| Do exercises!
| iueijasdkn wrote:
| I recommend "How to Think About Abstract Algebra" by Lara
| Alcock and "A Book of Abstract Algebra" by Charles Pinter.
| benrbray wrote:
| I second the book by Pinter -- the bulk of the book is a
| series of well-curated exercises meant to guide you through
| discovering the basic principles of abstract algebra for
| yourself. Excellent for self-study in the absence of a good
| professor.
| Vattazi wrote:
| If you need a solid theoretic reference, Jackobson's "Basic
| Algebra" books are recommended.
| chobytes wrote:
| That's definitely a good place to start. Having the right
| intuition for groups from the beginning makes everything much
| easier.
|
| As for other materials, what kind of math background do you
| have?
| graycat wrote:
| As a ugrad math major, I had some group theory. At the end, I
| wrote my honors paper on group representations in part to
| respond to a query from the chemistry department.
|
| I got the first dose of group theory in a standard course in
| abstract algebra that also covered Galois theory, rings,
| integral domains, fields, vector spaces, even a little on
| quaternions.
|
| In grad school I was forced into another course in abstract
| algebra, this time from the book I. Herstein, _Topics in
| Algebra_. Otherwise the prof was trying to understand algebraic
| geometry, as in Grothendieck. After his first few lectures on
| group theory, after class I asked if he was going to cover
| group representations, and he was intimidated and responded
| "That's deep". I explained that I'd just written my ugrad
| honors paper on that subject. So, I didn't bother to show up
| for his class, and at the end he gave me a little oral exam in
| Galois theory, which I had to review at the time.
|
| One good thing from this prof: He was Italian; the university
| was in a small town awash in dimly lit, romantic pizza shops;
| and some of us students asked him which pizza was better,
| Italian or American. He hesitated and confessed "American". So,
| he was both a mathematician and a diplomat!
|
| Apparently some people think that Herstein's book is still
| super good stuff: At
|
| https://www.alibris.com
|
| they want $36 for a used one and $200+ in some cases.
|
| I suspect that
|
| S. Lang, _Algebra_ is excellent although maybe not as a first
| or only source. I found a PDF at
| https://math24.files.wordpress.com/2013/02/algebra-serge-lan...
|
| Just the basics of group theory are so well known, written
| about so often that finding good sources should be easy. Say,
| just go to the section of any research library, see where
| Herstein's or Lang's books are, and see what else is there.
|
| Of course, here with Herstein and Lang I'm talking math,
| legitimately _pure math_ , and not computing, computer science,
| or much if anything in recent applications.
|
| The definition of a group and the early results are not just
| simple -- they are childishly, dirt simple, so simple that
| maybe an adult should be embarrassed to study them.
|
| After some decades, here from just memory I will guess at the
| definition: A _group_ is a non-empty set G together with an
| _operation_ (here I omit a picky definition of an _operation_
| ), say, + such that
|
| (i) For any elements a, b in G, a + b is in G (likely redundant
| given a careful definition of an _operation_ on G).
|
| (ii) There is an element 0, the _identity_ element, in G such
| that for any a in G we have
|
| a + 0 = 0 + a = a
|
| (iii) For any element a in G there exists an _inverse_ of a,
| -a, such that
|
| a + -a = -a + a = 0
|
| (iv) The operation + is _associative_ , that is, for any
| elements a, b, c in G, we have
|
| (a + b) + c = a + (b + c)
|
| So, let's check: Suppose a, b are in G and we have
|
| a + b = 0
|
| Then
|
| -a + (a + b) = -a
|
| Using associativity we have
|
| (-a + a) + b = -a
|
| Using the identity 0 we have
|
| 0 + b = -a
|
| and finally
|
| b = -a
|
| and similarly for
|
| b + a = 0
|
| So, the inverse of a is unique.
|
| How 'bout that! We have proved our first theorem in group
| theory!
|
| For a little lesson, note the extreme care here where we use
| associativity and the identity. In a lot of math, we get just
| to slop through such details, but in group theory we have to be
| _picky_ in the extreme.
|
| To continue with _group theory_ , we make another definition:
| Given group (G,+) (there is a style of math that likes such
| sparse, precise notation), if for any elements a, b in G, we
| have
|
| a + b = b + a
|
| then the group is _commutative_ and _Abelian_ (after the
| mathematician Abel).
|
| Not all groups are Abelian, that is, they are not all
| commutative. For a simple example, given quickly (I'm omitting
| some picky details) in matrix theory in linear algebra matrix
| addition is commutative but matrix multiplication is not.
|
| The topic _group representations_ replaces each element of
| group G with (usually, maybe always) a matrix and the operation
| with matrix multiplication. An advantage is that we get to
| write down the matrices and multiply them; otherwise we are
| likely stuck with a big group multiplication (or in the case of
| calling the operation +, addition) table.
|
| From all I can tell, the applications of group theory in
| quantum mechanics, particle physics, and chemistry make use of
| only the simplest parts of group theory.
|
| I have a bias: In math, first I want to see an application.
| That is, a recipe for rabbit stew starts out "First catch a
| rabbit." Then maybe I can get interested in learning some math
| or doing some research in math that I can use for the
| application. I confess: This is a radical approach to math.
| This approach was easy enough to anticipate for someone early
| in their career long flooded with math topics and trying to
| make money to support a wife, needing to be selective, etc.
|
| With this approach, I asked what were the applications of
| abstract algebra -- groups, rings, fields, etc. Not getting a
| good answer, I concluded that abstract algebra was abstract
| nonsense, tricky, picky stuff created for no good reason and
| returned to math _analysis_ and its applications to physics,
| engineering, etc.
|
| Well, surprising or not, that conclusion is wrong: Whatever
| Galois, Abel, etc. did/did not know about applications,
| abstract algebra got applied, has some applications. And
| abstract algebra is soon not childishly simple and, instead,
| has some deep/difficult questions and by now some
| deep/difficult answers.
|
| To continue on a little, a group G might have only finitely
| many elements or infinitely many elements. For a group with
| infinitely many elements -- super easy, with R the set of real
| numbers, (R,+) is an example.
|
| Given a group (G,+), maybe there is a proper subset H of G so
| that (H,+) (to be really picky, there is a slight abuse of
| notation here since an operation + cannot be the _same_ on both
| G and H -- group theory is a picky subject) is also a group.
| Then (H,+) is a _sub-group_ of (G,+).
|
| Exercise: Fine a proper subgroup of (R,+) (uh, to be picky,
| _proper_ here means not all of R).
|
| So, if we let |G| denote the number of elements in a set, we
| have a theorem: In case G is finite, |H| factors |G| where
| _factors_ means in the third grade sense divides with 0
| remainder. Exercise: Prove it!
|
| Some of the advanced results in group theory (Sylow's theorems,
| Jordan-Holder theory, and much more) are astounding -- tough to
| believe that they could be true, but they are.
| gradschool wrote:
| "For instance, the Galois group immediately tells you whether a
| polynomial can be solved at all ..."
|
| I wish she wrote one more sentence before changing the subject.
| If it's immediate, Can anyone explain the decision procedure?
| IngoBlechschmid wrote:
| The big picture is the following:
|
| There is a certain kind of groups called "solvable groups".
| There is a theorem stating that a polynomial equation can be
| solved by radicals if and only if its Galois group is a
| solvable group. And whether a given finite group (such as a
| Galois group) is solvable is an entirely finite combinatorial
| question which can be mechanically checked.
|
| Even better: A witness for the solvability of the Galois group
| can be transformed -- again nontrivially but mechanically --
| into explicit formulas for the solutions of the polynomial
| equation.
|
| Wikipedia has some examples on solvable groups and their
| connection to Galois theory, though I hope that someone has
| written some blog post or similar explaining this beautiful
| connection with less prerequisites.
| https://en.wikipedia.org/wiki/Solvable_group
| gizmo686 wrote:
| A polynomial is solvable if and only if its corresponding
| Galois group is "solvable". Solvable groups are motivating
| primarily by the problem of determining if polynomials are
| solvable. In general, it takes some work to know if a group is
| solvable, and I don't think there is a way to determine that
| immidiatly.
|
| The most charitable reading of the statement is that there are
| relativly few groups of small order. This means that for
| polynomials of relativly small degree you have probably seen
| their Galois group before so can go off of memory.
| tacomonstrous wrote:
| Not sure what this statement is supposed to mean. Every
| polynomial can be solved tautologically by adding a zero for it
| into the picture. For instance, every complex polynomial admits
| a complex zero (this is the Fundamental theorem of algebra), so
| you can find the zeros of any rational polynomial somewhere.
| You can of course look at the smallest 'field' in which these
| zeros live: basically, look at everything you get by writing
| down all possible arithmetic expressions involving those zeros.
| For the polynomial x^2-2, you get numbers of the form
| a+\sqrt(2)b where a and b are rational numbers.
|
| The Galois group tells you how complicated this new field is,
| and how involved its symmetries are. In the example above there
| is only one nontrivial symmetry which switches a+sqrt(2)b with
| a-sqrt(2)b. In general there are many more, and when you
| consider the symmetries of the zeros of all possible rational
| polynomials, you get the absolute Galois group of the rational
| numbers. It can be said without exaggeration that number theory
| is the study of this Galois group, which is still a rather
| mysterious object in its full glory.
| xyzzyz wrote:
| The intended (and, in fact, widely understood within
| mathematical community) meaning of that statement is whether
| the roots of polynomials can be expressed using radicals, ie.
| as algebraic expressions involving four standard arithmetic
| operations and taking roots.
|
| I'd rather quibble with the part where they say that this is
| "immediately" seen from Galois group, as showing solvability
| of a group is not trivial matter at all.
| voldacar wrote:
| Color me impressed! A Quanta article with an equation.
| alanbernstein wrote:
| The Abel-Ruffini theorem, that general quintics and higher have
| no solutions via radicals, feels to me like such a fundamental
| result, that there must be _some_ intuitive explanation for it. I
| have tried to understand the topic at a level that shows me that
| intuition, but I have never quite reached it. Any suggestions?
| anon_tor_12345 wrote:
| arnold's proof is the most intuitive and what made it click
| after years of being responsible for understanding it (several
| years of algebra at the undergrad and grad level):
|
| https://www.youtube.com/watch?v=zeRXVL6qPk4
| mjreacher wrote:
| Is there any difference between that video and this one?:
| https://www.youtube.com/watch?v=RhpVSV6iCko seems to be the
| same author but that video is 10 minutes longer.
| anon_tor_12345 wrote:
| there is some slight difference but i don't remember what
| it is. i wrote a blog post about abel-ruffini based on
| those two videos and i remember there being some things
| that were clearer in one rather than the other (but i don't
| remember which!)
| gmadsen wrote:
| I think the intuition only comes from understanding the
| arguments intuitively. so basically an entire algebra class,
| you need to understand field extensions and PIDs.
| paulpauper wrote:
| The original proofs are quite long. Not exacyly something you
| can compress to a comment or an article. I dunno
| go_elmo wrote:
| The proof reduces it to galois-groups and shows some easy
| properties, which are intuitive in the galois-group domain
| (only divisor of S_5 is A_5 which is not of prime-order). Maybe
| studying the relationship / reduction to / from polynomial to
| galois will help
| kortex wrote:
| Gonna butcher this, but it's good practice, so I'm gonna take a
| stab at it. The roots of polynomials are intimately related to
| the Alternating symmetry groups. These describe the ways you
| can permute a set, specifially A_n describes the even
| permutations of a set of n elements. The even permutations of a
| sequence are those obtained by making an even number of
| transpositions ("moves").
|
| I can make 2 swaps to go 1,2,3,4,5 -> 1,4,3,2,5 -> 1,4,5,2,3.
| That's an even permutation.
|
| The order (size) of A_n is the number of even permutations of n
| unique elements. The orders are as follows:
|
| A2: 1
|
| A3: 3
|
| A4: 12
|
| A5: 60
|
| A6: 360
|
| A_n: n! / 2
|
| The "solvability" of a polynomial depends on the the degrees of
| freedom and number of constraints. Quintic is where the number
| of degrees of freedom blows up and you can no longer factor out
| roots in such a way that you can uniquely factor the
| polynomial. Quadratic is basically trivial. Cubic cannot be
| directly factored, because you have 3 variables and 2
| equations, but you can "factor out" a quadratic that you can
| solve, and then solve the cubic. Quartic is the same, you end
| up with 4 variables and 3 unique equations, you can factor out
| a cubic, which you can solve, etc. Quartic, you have 4
| equations to solve and...something like 30 variables (or maybe
| it's 60. It's some way more than we can factor!).
|
| https://en.wikipedia.org/wiki/Alternating_group
|
| https://math.stackexchange.com/questions/550401/intuitive-re...
| paulpauper wrote:
| Sorta. There are obviously infinite quintics that can be
| solved despite this. For example any polynomial in which the
| roots are arranged on a circle on the real,complex plane
| jstx1 wrote:
| It's apparently the topic of 3blue1brown's next video -
| https://twitter.com/3blue1brown/status/1420461376896520193
| jordigh wrote:
| If you're into this stuff, I really like this book on Galois
| theory:
|
| https://www.worldcat.org/title/classical-introduction-to-gal...
|
| The reason I like this book is that it is very computational, in
| the spirit of Galois's original discoveries. Modern treatment of
| Galois theory has been greatly coloured by Emil Artin's
| reformulation of the subject: the Galois group is nowadays
| primarily taught to undergrads as being a group of automorphisms
| of a field extension instead of being primarily the computational
| object of the permutations of the roots of a polynomial.
|
| Artin's approach has merit, of course. The extra level of
| abstraction generalises to much more than just simple algebraic
| extensions of the rationals. However, I think the historical
| approach in this case is more instructive. Seeing lots of
| calculations before the abstractions gives you a better sense of
| what's being abstracted away.
| dmch-1 wrote:
| There is Artin's approach, and then there is a more abstract
| approach by Grothendieck relating to the fundamental group of
| algebraic topology.
|
| In the movie Beautiful mind there is a scene, where a student
| tells John Nash that he can proof that 'Galois extensions are
| the same as covering spaces'. This follows from the
| Grothendieck's approach. However the analogy between Galois
| extessions and the fundamental group was known even before
| Grothendieck.
|
| Then there are even more general approaches in the category
| theory setting.
|
| Today these generalisations are taught indeed without much
| regard to the computational spirit of 19th century mathematics.
| They have their merit as you say, but I agree understanding the
| computational aspects are instructive in fully appreciating the
| generalisations and analogies.
| cmpb wrote:
| Reaching Galois groups (and field extensions, rings, and other
| related tools) in class was the point in my math undergrad that I
| realized that math had become too subtle for my brain to easily
| comprehend. I quickly realized that I'd either have to start
| studying outside of class to learn the motivations and
| connections or accept that I would not be pursuing math as a
| career.
|
| And that's when I got a job in software.
| nobody0 wrote:
| You can still porting those algebraic structures to day to day
| software construction.
|
| https://corecursive.com/050-sam-ritchie-portal-abstractions-...
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