[HN Gopher] Galois Theory
___________________________________________________________________
Galois Theory
Author : mathgenius
Score : 305 points
Date : 2024-08-15 13:00 UTC (9 hours ago)
(HTM) web link (golem.ph.utexas.edu)
(TXT) w3m dump (golem.ph.utexas.edu)
| 082349872349872 wrote:
| In particular, abuse of Galois Theory makes it possible to
| reconcile Spinoza with Aquinas.
| esafak wrote:
| Go on.
| koolala wrote:
| Reconcile Liebniz too. I agree this sounds awesome.
| 082349872349872 wrote:
| 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 wrote:
| ;) A mention is enough. Monads are LIFE!
| 082349872349872 wrote:
| 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, m, e) exist? what about (G,
| d, [?]) in the other direction? If they do exist, what
| are they, for Aquinas, Leibniz, and Spinoza?
| novosel wrote:
| Why would they need to be reconciled? Or, indeed, Galoas
| abused?
| 082349872349872 wrote:
| 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 , 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 wrote:
| 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.
| rpmw wrote:
| 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
| mcfunley wrote:
| 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 wrote:
| 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 wrote:
| 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.
| agumonkey wrote:
| 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 wrote:
| 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 wrote:
| 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
|
| 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.
| koolala wrote:
| Did the solution taking 10 minutes make it seem like it was
| all just semantics from old faulty definitions?
|
| How do you personally imagine trisecting an angle now? Is it
| possible to describe your new intuition of the impossibility
| in different human understandable terms that are also
| geometric? Impossible things are weird conversation subjects.
| openasocket wrote:
| > Did the solution taking 10 minutes make it seem like it
| was all just semantics from old faulty definitions?
|
| I recall that I, and the rest of the class, were very
| suspicious of the proof. The proof took maybe 10 minutes,
| but it probably took another 10-15 minutes for the
| professor to convince us there wasn't a logical error in
| the given proof. Though the situation was kind of the
| opposite of what you would thing. We understood field
| extensions and the symmetries of the roots of polynomials
| really well. What took convincing was that any formula
| using addition, subtraction, multiplication, division,
| exponents, and rational roots would always give you a field
| extension that mapped to a "solvable group". The proof is
| essentially:
|
| 1. Any field extension of a number constructed using those
| mathematical operations must map to a solvable group. 2.
| For every group there exists a corresponding field
| extension (this is a consequence of the fundamental theorem
| of Galois theory). 3. There exist groups that are not
| solvable. 4. Therefore, there are polynomials with roots
| that can't be constructed from the elementary mathematical
| operations.
|
| Basically the entire course is dedicated to laying out part
| 3, and the part we were suspicious about was part 1.
|
| The one thing that is interesting about the proof is that
| it is actually partially constructive. Because there is no
| general quintic formula, but there are some quintics that
| are solvable. For instance, x^5 - 1 clearly has root x=1.
| And Galois theory allows you to tell the difference between
| those that are solvable and those that are not. It allows
| you to take any polynomial and calculate the group of
| symmetries of those roots. If that group is solvable, then
| all of the roots can be defined in terms of elementary
| operations. If not, at least one of the roots cannot.
|
| > How do you personally imagine trisecting an angle now? Is
| it possible to describe your new intuition of the
| impossibility in different human understandable terms that
| are also geometric?
|
| So the trisection proof I don't remember as well, but
| looking it up it isn't very geometric. It essentially
| proves that trisecting an angle with a compass and straight
| edge is equivalent to solving certain polynomial equations
| with certain operations, and goes into algebra.
|
| That said, Galois theory itself feels very "geometric" in
| the roughest sense of the term. Fundamentally, it's about
| classifying the symmetries of an object.
| kevinventullo wrote:
| _2. For every group there exists a corresponding field
| extension (this is a consequence of the fundamental
| theorem of Galois theory)._
|
| Just a nit, but when talking about extensions of Q, this
| is called the Inverse Galois Problem and it is still an
| open problem.
|
| That said, you don't actually need this strong of a
| statement to show general insolvability of the quintic.
| Rather you just need to exhibit a _single_ extension of Q
| with non-solvable Galois group. I believe adjoining the
| roots of something like x^5+x+2 suffices.
| koolala wrote:
| Your story makes me picture Geometric Algebra, defining
| Complex Numbers or Quaternions and multiplication on
| them, and their symmetries and elegant combinations. Ty
| for sharing your math memories. Makes me wonder if Galois
| theory can determine valid or invalid imaginary number
| combinations / systems.
| ForOldHack wrote:
| "the proof of impossibility of trisecting an angle."
|
| It makes it into a silly little footnote, a very little
| footnote, I was both sad and disappointed when I read and
| understood it in an appendix.
|
| Now back to Pi + e = Pie.
| agumonkey wrote:
| Perspective change are wonderful. Sometimes you realize late in
| life that people from 100+ years ago were so advanced
| intellectually.
| 082349872349872 wrote:
| On the other hand, getting yourself shot in the gut 152 years
| before the invention of the Bogota bag might be a textbook
| example of the english chengyu "book smart life stupid"?
|
| (see
| https://vuir.vu.edu.au/18204/1/PETSINIS_1995compressed.pdf
| but note that as well as being a fictional account, Petsinis
| has made some factual errors in details taken from the
| coroner's report:
| https://news.ycombinator.com/item?id=40650555 )
| VyseofArcadia wrote:
| 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 wrote:
| 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 wrote:
| 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 Dieudonne
| 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.
| wging wrote:
| Do you know how Dieudonne answered?
| Koshkin wrote:
| _On foundations we believe in the reality of mathematics,
| but of course, when philosophers attack us with their
| paradoxes, we rush to hide behind formalism and say
| 'mathematics is just a combination of meaningless
| symbols,'... Finally we are left in peace to go back to
| our mathematics and do it as we have always done, with
| the feeling each mathematician has that he is working
| with something real. The sensation is probably an
| illusion, but it is very convenient._
| stoneman24 wrote:
| The 'make thier own mental models' vs sharing/providing
| full information is difficult.
|
| 'Make thier own model' of the domain can lead to deeper
| understanding but takes time and may lead to different
| (possibly incorrect) understanding of the issues and
| complexities. If not reviewed with others.
|
| Providing full information upfront to a person can be
| quicker but lead to a superficial knowledge.
|
| I think that it comes down to whether that deeper knowledge
| is directly needed for the main task. Can I get by with an
| superficial (leaky) abstraction and concentrate on the main
| job.
| fredilo wrote:
| > 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.
|
| If the objective is to advance mathematics instead of
| making it accessible, then this is a somewhat reasonable
| position. The mathematical statements that a person can
| come up with is often a direct product of their mental
| image. If everyone has the same image, everyone comes up
| with similar mathematical statements. For this reason you
| want to avoid that everyone has the same picture. Forcing
| everyone to start with a clean canvas increases the chance
| that there is diversity in the images. Maybe someone finds
| a new image, that leads to new mathematical statements. At
| least that's the idea. One could also argue that it just
| leads to blank canvases everywhere.
| paulddraper wrote:
| Elements is formal, right?
| hollerith wrote:
| Pretty much. According to someone on Stack Exchange, "The
| most serious difficulties with Euclid from the modern point
| of view is that he did not realize that an axiom was needed
| for congruence of triangles, Euclids proof by superposition
| is not considered as a valid proof." But making mistakes in
| a formal treatment of a subject does not negate the fact
| that it is a formal treatment of the subject IMHO, and
| AFAICR none of the theorems in Elements is wrong; i.e.,
| Elements is formal enough to have avoided a mistake even
| though the axioms listed weren't all the axiom that are
| actually needed to support the theorems.
|
| 18th Century European math was much more potent than
| ancient Greek math, and although parts of it like algebra
| and geometry were, for a long time, most of it was not
| understood at a formal or rigorous level for a long time
| even if we accept the level of rigor found in Elements.
| koolala wrote:
| Isn't how formal or rigorous something is just a social
| convention? Grammer Nazi's used to make online speech be
| formal with perfect rigor. Isn't it all relative to what
| your society defines?
| chongli wrote:
| No. That's the colloquial definition of formal. In
| mathematics, the word formal refers to something more
| specific: one or more statements written using a set of
| symbols which have fully-defined rules for mechanically
| transforming them into another form.
|
| A formal proof is then one which proceeds by a series of
| these mechanical steps beginning with one or more
| premises and ending with a conclusion (or goal).
| koolala wrote:
| Both are a formality based on whats in fashion. I like
| the Axiom of Choice and not taking Math or words as
| literal or biblical truth.
| chongli wrote:
| If you're a formalist in philosophy of math, then math is
| neither true nor false, it's merely a bunch of
| meaningless symbols you transform via mechanical rules.
| nyssos wrote:
| To an extent. A truly completely _formal_ proof, as in
| symbol manipulation according to the rules of some formal
| system, no. It 's valid or it isn't.
|
| But no one actually works like this. There are varying
| degrees of "semiformality" and what is and isn't
| acceptable is ultimately a convention, and varies between
| subfields - but even the laxest mathematicians are still
| about as careful as the most rigorous physicists.
| gowld wrote:
| Elements is mostly formal, but it's also concrete and
| visual.
|
| Euclid developed arithmetic and algebra through
| constructive geometry, which relies on our visual intuition
| to solve problems. Non-concrete problems were totally out
| of scope. Even curved surfaces (denying the parallel
| postulate) were byond Euclid. Notably, Elements didn't have
| imaginary or transcendental numbers. Euclid made no attempt
| to unify line lengths and arc lengths, and had nothing to
| say about what fills the gaps between the algebraically
| (geometrically!) constructible numbers.
| paulddraper wrote:
| > Elements didn't have imaginary or transcendental
| numbers
|
| Elements has pi.*
|
| *It proves the ratio of a circle's area to the square of
| its diameter is constant.
| gowld wrote:
| But doesn't say how large that constant value is.
| Someone wrote:
| > Euclid [...] had nothing to say about what fills the
| gaps between the algebraically (geometrically!)
| constructible numbers
|
| Did he even know those gaps existed? Euclid lived around
| 300BC. The problem of squaring the circle had been
| proposed around two centuries before that
| (https://en.wikipedia.org/wiki/Anaxagoras#Mathematics),
| but I don't think people even considered it to be
| impossible by that time.
| Koshkin wrote:
| Incidentally, here's one beautiful edition:
|
| https://farside.ph.utexas.edu/Books/Euclid/Elements.pdf
| dataflow wrote:
| Do they explain anywhere why we consider radicals special and
| ask these questions about them in the first place? Nobody ever
| explains that. To my programmer brain, whether I'm solving t^3
| = 2 or t^100 + t + 1 = 7 numerically, I have to use an
| iterative method like Newton's either way. ("But there's a
| button for Nth root" isn't an argument here - they could've
| added a more generic button for Newton.) They don't
| fundamentally seem any different. Why do I care whether they're
| solvable in terms of _radicals_? It almost feels as arbitrary
| as asking whether something is solvable without using the digit
| 2. I would 've thought I'd be more interested in whether
| they're solvable in some other respect (via iteration, via
| Newton, with quadratic convergence, or whatever)?
| Joker_vD wrote:
| I imagine it has something to do with the fact that there is
| a somewhat simple pen-and-paper procedure, quite similar to
| long division, for calculating square/cubic/etc. roots to
| arbitrary precision, digit by digit. So finding roots is, in
| a sense, an arithmetic operation; and of calculus one better
| not speak in a polite society.
| phs wrote:
| Radicals are a "natural" extension in a certain sense, just
| as subtraction and division are. They invert an algebraic
| operation we often encounter when trying to solve equations
| handed to us by e.g. physics. I find it understandable to
| want to give them a name.
|
| Why not something like "those things we can solve with
| Newton"? As you note Newton is broadly applicable; one would
| hope, given how popular the need to invert an exponent is,
| that something better (faster, more stable) if more specific
| than Newton might be created. It is hard to study a desired
| hypothetical operation without giving it a name.
|
| On a related note, how come we don't all already have the
| names of the 4th order iterative operation (iterated
| exponents) and its inverse in our heads? Don't they deserve
| consideration? Perhaps, but nature doesn't seem to hand us
| instances of those operations very often. We seemingly don't
| need them to build a bridge or solve some other common
| practical engineering problem. I imagine that is why they
| fail to appear in high school algebra courses.
| bubble12345 wrote:
| You are right in the sense that solvability by radicals has
| no practical importance, especially when it comes to
| calculations.
|
| It is just a very classical pure math question, dating back
| hundreds of years ago. Its solution led to the development of
| group theory and Galois theory.
|
| Group theory and Galois theory then are foundational in all
| kinds of areas.
|
| Anyway, so why care about solvability by radicals? To me the
| only real reason is that it's an interesting and a natural
| question in mathematics. Is there a general formula to solve
| polynomials, like the quadratic formula? The answer is no -
| why? When can we solve a polynomial in radicals and how?
|
| And so on. If you like pure math, you might find solvability
| by radicals interesting. It's also a good starting point and
| motivation for learning Galois theory.
| VyseofArcadia wrote:
| > To my programmer brain, whether I'm solving t^3 = 2 or
| t^100 + t + 1 = 7 numerically
|
| Typically when you're solving a polynomial equation in an
| applied context (programming, engineering, physics,
| whatever), it's because you have modeled a situation as a
| polynomial, and the information you really want is squirreled
| away as the roots of that polynomial. You don't actually care
| about the exact answer. You've only got k bits of precision
| anyway, so Newton's method is fine.
|
| But we're not interested in the solution. We don't
| particularly care that t = approx. 1.016 is a numerical
| solution.[0] We're not using polynomials to model a
| situation. In mathematics, we are often studying polynomials
| as objects in and of themselves, in which case the kind of
| roots we get tells us something about polynomials work, or we
| are using polynomials as a lens through which to study
| something else. In either case it's less about a specific
| solution, and more about what kind of solution it is and how
| we got it.
|
| Not to mention, specific polynomials are examples. Instead of
| t^100 + t + 1 = 7, we're usually looking at something more
| abstract like at^100 + bt + c.
|
| [0] And in the rare case we actually care about a specific
| root of a specific polynomial, and approximate numerical
| solution is often not good enough.
| photonthug wrote:
| > Do they explain anywhere why we consider radicals special
| and ask these questions about them in the first place? Nobody
| ever explains that. To my programmer brain,
|
| But regardless of whether you're a programmer, mathematician,
| machinist, carpenter, or just a kid playing with legos,
| there's _always_ a good time to be had in the following way:
| first you look at the most complex problems that you can
| manage to solve with simple tools; then you ask if your
| simple tools are indeed the _simplest_ ; and then if multiple
| roughly equivalently simple things are looking tied in this
| game you've invented then now you get the joy of endlessly
| arguing about what is most "natural" or "beautiful" or what
| "simple" even means _really_. Even when this game seems
| pretty dumb and arbitrary, you 're probably learning a lot,
| because even when you can't yet define what you mean by
| "simple" or "natural" or "pretty" it's often still a useful
| search heuristic.
|
| What can you do with a lot of time and a compass and a ruler?
| Yes but do we need the "rule" part or only the straight-edge?
| What can we make with only SKI combinators? Yes but how
| awkward, I rather prefer BCKW. Who's up for a round of code-
| golf with weird tiny esolangs? Can we make a zero
| instruction-set computer? What's the smallest number of tools
| required to put an engine together? Yes but is it
| theoretically possible that we might need a different number
| of different tools to take one apart? Sure but does that
| really count as a separate tool? And so it goes.. aren't we
| having fun yet??
| impendia wrote:
| I'd say one of the fundamental lessons of field and Galois
| theory is that they're _not_ intrinsically special. They 're
| just easier and more appealing to write down. (For the most
| part, anyway. They're a little bit special in some
| subspecialties for some technical reasons that are hard to
| explain.)
|
| One reason to focus on them: when you tell students that x^5
| - x - 1 = 0 can't be solved by radicals, that "even if God
| told you the answer, you would have no way to write it down",
| this is easy to understand and a powerful motivator for the
| theory. It's a nice application which is not fundamental, but
| which definitely shows that the theory has legs.
|
| If you want to know which polynomials are solvable by
| Newton's method? All of them. It illustrates that Newton's
| method is extremely useful, but the answer itself is not
| exactly interesting.
| returningfory2 wrote:
| > "even if God told you the answer, you would have no way
| to write it down"
|
| But isn't this also true for generic quadratics/cubics too?
| Like the solution to x^3-2=0 is cubed_root_of(2), so it
| seems we can "write it down". But what is the definition of
| cubed_root_of(2)? Well, it's the positive solution to
| x^3-2=0...
| impendia wrote:
| When I say that a fundamental lesson of field theory is
| that radicals are not really special, this is what I
| mean. You are thinking in a more sophisticated way than
| most newcomers to the subject.
| returningfory2 wrote:
| Oh sorry, right.
|
| I feel there is an interesting follow-up problem here.
| The polynomials x^n+a=0 are used to define the "radicals"
| which is a family of functions F_n such that F_n(a) =
| real nth root of a = real solution of x^n + a = 0. Using
| these radicals you can solve all quadratics, cubics and
| quintics.
|
| Now take another collection of unsolvable polynomials;
| your example was x^5 - x - 1 = 0 and maybe parameterize
| that in some way such that these polynomials are
| unsolvable. This gives us another family of functions
| G_n. What if we allow the G_n's to be used in our
| solutions? Can we solve all quintics this way (for
| example)?
| donkeybeer wrote:
| Bring radicals
| returningfory2 wrote:
| Awesome reference, thank you!
| impendia wrote:
| I'm not sure I understand your question exactly, but I am
| fairly certain that not all quintic fields can be solved
| by the combination of (1) radicals, i.e. taking roots of
| x^n - 1, and (2) taking roots of x^5 - x - 1. I don't
| have a proof in mind at the moment, but I speculate it's
| not too terribly difficult to prove.
|
| If I'm correct, then the proof would almost certainly use
| Galois theory!
| returningfory2 wrote:
| Thanks! I should have made it more explicit that we would
| need some family of quantic equations, not just x^5 - x -
| 1. And looks like from another reply the answer is yes? h
| ttps://en.wikipedia.org/wiki/Bring_radical#Solution_of_th
| e_...
| itishappy wrote:
| > Can we solve all quintics this way (for example)?
|
| Nope. Wanna hazard a guess what theory was instrumental
| in proving that there's no closed form for general
| quintics?
|
| https://en.wikipedia.org/wiki/Galois_theory#A_non-
| solvable_q...
| donkeybeer wrote:
| He is not wholly wrong, while not solvable in the general
| case by normal radicals, there is a family of functions,
| a special "radical" as he said, the Bring radical that
| solves the quintic generally. Of course as said its not a
| 5th root, but the solution to a certain family of
| quintics.
| itishappy wrote:
| Fascinating, thanks!
| enugu wrote:
| The prime importance of solving by radicals is actually, that
| it led to the theory of groups! Groups are used in all sorts
| of places. (One nice pictorial example is fundamental group
| of a topological space). Just like Complex numbers arose in
| trying to solve the cubic. Also, the statement of Fermat's
| Last Theorem doesn't have any applications but its solution
| led to lot of interesting theory like how ideals get
| factorized in rings, elliptic curves, Galois
| representations...
|
| BTW, the same theory can be extended to differential
| equations and Differential Galois Theory tells you if you can
| get a solution by composing basic functions along with
| exponentials.
|
| Historically, radicals can be motivated by looking at people
| trying to solve linear, then quadratics, medieval duels about
| cubics and quartics, the futile search for solving quintics
| etc. Incidentally, quintics and any degree can have a closed
| form solution using modular functions.
|
| More discussion on the MathOverflow page
| https://mathoverflow.net/questions/413468/why-do-we-make-
| suc...
| fredilo wrote:
| Superb question! Interestingly, my math professor asked the
| exact same question after teaching Galois theory and stated
| that he does not have a good answer himself. Let me try to
| give sort of an answer. :)
|
| We have fingers. These we can count. This is why we are
| interested in counting. This is gives us the natural numbers
| and why we are interested in them. What can we do with
| natural numbers? Well the basic axioms allow only one thing:
| Increment them.
|
| Now, it is a natural question to ask what happens when we
| increment repeatedly. This leads to addition of natural
| numbers. The next question is to ask is whether we can undo
| addition. This leads to subtraction. Next, we ask whether all
| natural numbers can be subtracted. The answer is no. Can we
| extend the natural numbers such that this is possible? Yes,
| and in come the integers.
|
| Now, that we have addition. We can ask whether we can repeat
| it. This leads to multiplication with a natural number. Next,
| we ask whether we can undo it and get division and rational
| numbers. We can also ask whether multiplication makes sense
| when both operands are non-natural.
|
| Now, that we have multiplication, we can ask whether we can
| repeat it. This gives us the raising to the power of a
| natural number. Can we undo this? This gives radicals. Can we
| take the root of any rational number? No, and in come
| rational field extensions including the complex numbers.
|
| A different train of thought asks what we can do with mixing
| multiplication and addition. An infinite number of these
| operations seems strange, so let's just ask what happens when
| we have finite number. It turns out, no matter how you
| combine multiplication and addition, you can always rearrange
| them to get a polynomial. Formulated differently: Every
| branch-free and loop-free finite program is a polynomial
| (when disregarding numeric stability). This view as a program
| is what motivates the study of polynomials.
|
| Now, that we have polynomials, we can ask whether we can undo
| them. This motivates looking at roots of polynomials.
|
| Now, we have radicals and roots of polynomials. Both
| motivated independently. It is natural to ask whether both
| trains of thought lead to the same mathematical object.
| Galois theory answers this and says no.
|
| This is a somewhat surprising result, because up to now, no
| matter in which order we asked the questions: Can we repeat?
| Can we undo? How to enable undo by extension? We always ended
| up with the same mathematical object. Here this is not the
| case. This is why the result of Galois theory is so
| surprising to some.
|
| Slightly off-topic but equally interesting is the question
| about what happens when we allow loops in our programs with
| multiplication and addition? i.e. we ask what happens when we
| mix an infinite number of addition and multiplication. Well,
| this is somewhat harder to formalize but a natural way to
| look at it is to say that we have some variable, in the
| programming sense, that we track in each loop iteration. The
| values that this variable takes forms a sequence. Now, the
| question is what will this variable end up being when we
| iterate very often. This leads to the concept of limit of a
| sequence.
|
| Sidenote: You can look at the usual mathematical limit
| notation as a program. The limit sign is the while-condition
| of the loop and the part that describes the sequence is the
| body of the loop.
|
| Now that we have limits and rational numbers, we can ask how
| to extend the rational numbers such that every rational
| sequence has a limit. This gives us the real numbers.
|
| Now we can ask the question of undoing the limit operation.
| Here the question is what undoing here actually means. One
| way to look at it is whether you can find for every limit,
| i.e., every real number, a multiply-add-loop-program that
| describes the sequence whose limit was taken. The answer
| turns out to be no. There is a countable infinite number of
| programs but uncountably infinite many real numbers. There
| are way more real numbers than programs. In my opinion this
| is a way stranger result than that of Galois theory. It turns
| out, that nearly no real number can be described by a
| program, or even more generally any textual description. For
| this reason, in my opinion, real numbers are the strangest
| construct in all of mathematics.
|
| I hope you found my rambling interesting. I just love to talk
| about this sort of stuff. :)
| glompers wrote:
| Yep!
| dataflow wrote:
| Thanks so much, I feel like you're the only one who grasps
| the crux of my question!
|
| > This is a somewhat surprising result, because up to now,
| no matter in which order we asked the questions: Can we
| repeat? Can we undo? How to enable undo by extension? We
| always ended up with the same mathematical object.
|
| I think this is the bit I'm confused on - we have an
| operation that is a mixture of two operations, where
| previously we only looked at pure compositions of
| operations (let's call this "impure"). Why is it surprising
| that the inversion of an "impure" operation produces an
| "impure" value?
|
| It's like saying, if I add x to itself a bunch, I always
| get a multiple of x. If I do the same thing with y, then I
| get a multiple of y. But if I add x and y to each other, I
| might get a prime number! Is that surprising? Mixing is a
| fundamentally new kind of operation; why would you expect
| its inversion to be familiar?
| lanstin wrote:
| They aren't really special except that adjoining the
| solutions to a radical to a field make the associated group
| of automorphisms simplify in a nice way. Also at the time
| tables of radical roots where common technology and Newton's
| method was not. But fundamentally, we can define and use new
| functions to solve things pretty easily; if you get down to
| it cos() and sin() are an example of this happening. All
| those applied maths weird functions (Bessel, Gamma, etc) are
| also this. As well, the reason not to just use numeric
| solutions for everything is that there are structural or
| dynamic properties of the solutions of equations that you
| won't be able to understand from a purely numeric
| perspective.
|
| I think taking a specific un-solvable numeric equation and
| deriving useful qualitative characteristics is a useful thing
| to try. You have cool simple results like Lyapunov stability
| criterion or signs of the eigenvalues around a singularity,
| and can numerically determine that a system of equations will
| have such and such long term behavior (or the tests can be
| inconclusive because the numerical values are just on the
| threshold between different behaviors.
|
| That's one of the really fun things about taking Galois
| theory class - you get general results for "all quintics" or
| "all quadratics" but also you can take specific polynomials
| and (sometimes) get concrete results (solvable by radicals,
| but also complex vs. real roots, etc.).
| sesm wrote:
| In a real problem you can have equations with parameters,
| where coefficients depend on some other parameters. And you
| should be able to answer questions about the roots depending
| on parameter values, like 'in which range of parameters p you
| have real (non-complex) roots'? Having a formula for the
| roots based on coefficients lets you answer such questions
| easily. For example, for quadratic equation ax^2 +bx + c = 0
| we have D = b^2 - 4ac, and if D < 0 then there is no real
| (non-complex) roots.
| nicf wrote:
| Lots of good answers here already, but I can also add my own
| perspective. Fundamentally, you're right that there isn't
| really anything "special" about radicals. The reason I
| personally find the unsolvability of the quintic by radicals
| interesting is that you _can_ solve quadratics, cubics, and
| quatrics (that is, polynomial equations of degree 2, 3, and
| 4) by radicals.
|
| To say the same thing another way: the quadratic formula that
| you learned in high school has been known in some form for
| millennia, and in particular you can reduce the question of
| solving quadratics to the question of finding square roots.
| So (provided you find solving polynomial equations to be an
| interesting question) it's fairly natural to ask whether
| there's an analogous formula for cubics. And it turns out
| there is! You need both cube roots and square roots, and the
| formula is longer and uglier, but that's probably not
| surprising.
|
| Whether you think n'th roots are "intrinsically" more
| interesting than general polynomial equations or not, this is
| still a pretty striking pattern, and one might naturally be
| curious about whether it continues for higher degrees. And I
| don't know anyone whose first guess would have been "yes, but
| only one more time, and then for degree 5 and higher it's
| suddenly impossible"!
| Koshkin wrote:
| I think it's quite natural to ask oneself how far one can go
| applying the same method that is used for solving the
| equation x2 = 4.
| Someone wrote:
| > To my programmer brain, whether I'm solving t^3 = 2 or
| t^100 + t + 1 = 7 numerically, I have to use an iterative
| method like Newton's either way.
|
| Doesn't your programmer brain want things to run as fast as
| possible?
|
| If you have another weapon in your arsenal for solving
| polynomial equations, you have an extra option for improving
| performance. As a trivial example, you don't call your Newton
| solver to solve a linear equation, as the function call
| overhead would mean giving up lots of performance.
|
| Also, if you solve an equation not because you want to know
| the roots of the equation but because you want to know
| whether they're different, the numerical approach may be much
| harder than the analytical one.
|
| > I would've thought I'd be more interested in whether
| they're solvable in some other respect (via iteration, via
| Newton, with quadratic convergence)
|
| That's fine. For that, you read up on the theory behind
| various iterative methods.
| atribecalledqst wrote:
| On the subject of more context in math, I've always wondered if
| having a grasp of the history of math would be helpful in
| getting better at solving mathematical problems. i.e. would
| learning more about how math developed over time, and how
| people solved important problems in the past, help me in trying
| to solve some other problem today?
|
| Years ago I bought the 3-volume set "Mathematical Thought from
| Ancient to Modern Times", but never had the time to get past
| the first few chapters. I'd be interested in any
| recommendations for math history tomes like that.
| mbivert wrote:
| > I'd be interested in any recommendations for math history
| tomes like that.
|
| Not a book, but FWIW, I've enjoyed a few videos from Norman
| Wildberger's "Math History" playlist[0]. Interestingly, he
| has a unconventional view of infinite processes in
| mathematics, a point of view that used to be common about a
| century ago or so.
|
| I'm sure knowing some amount of history is useful, but there
| must be a limit to how much of it is practically useful
| though.
|
| https://www.youtube.com/watch?v=dW8Cy6WrO94&list=PL55C7C8378.
| ..
| mindcrime wrote:
| I've been going through that very video lecture series the
| last couple of weeks. Good stuff. And in the lectures he
| mentions a number of books. I looked a few up on Amazon,
| and then looked at the associated Amazon recommendations,
| and so far have this small list of books related to Maths
| history that look worth reading:
| Mathematics and Its History (Undergraduate Texts in
| Mathematics) 3rd ed. 2010 Edition by John Stillwell
| The History of the Calculus and Its Conceptual Development
| (Dover Books on Mathematics) by Carl B. Boyer A
| History of Mathematics by Carl B. Boyer A
| Concise History of Mathematics: Fourth Revised Edition
| (Dover Books on Mathematics)A Concise History of
| Mathematics: Fourth Revised Edition (Dover Books on
| Mathematics) by Dirk J. Struik Introduction to
| the Foundations of Mathematics: Second Edition (Dover Books
| on Mathematics) Second Edition by Raymond L. Wilder
| Mathematical Thought from Ancient to Modern Times by Morris
| Kline (3 volume set) The Calculus Wars: Newton,
| Leibniz, and the Greatest Mathematical Clash of All Time by
| Jason Socrates Bardi
|
| There is also a "thing" in mathematics that is sometimes
| called the "genetic approach" where "genetic" is roughly
| equivalent to "historical" or maybe "developmental". IOW, a
| "genetic approach" book teaches a subject by tracing the
| development of the subject over its history. One popular
| book in this mold is: The Calculus: A
| Genetic Approach by Otto Toeplitz
| nyssos wrote:
| You're conflating a few things here.
|
| Constructivists are only interested in constructive proofs:
| if you want to claim "forall x in X, P(x) is true" then you
| need to exhibit a particular element of x for which P
| holds. As a philosophical stance this isn't super rare but
| I don't know if I would say it's ever been common. As a
| field of study it's quite valuable.
|
| Finitists go further and refuse to admit any infinite
| objects at all. This has always been pretty rare, and it's
| effectively dead now after the failure of Hilbert's
| program. It turns out you lose a ton of math this way -
| even statements that superficially appear to deal only with
| finite objects - including things as elementary as parts of
| arithmetic. Nonetheless there are still a few serious
| finitists.
|
| Ultrafinitists refuse to admit any _sufficiently large
| finite objects_. So for instance they deny that
| exponentiation is always well-defined. This is completely
| unworkable. It 's ultrafringe and always has been.
|
| Wildberger is an ultrafinitist.
| mbivert wrote:
| > You're conflating a few things here.
|
| It's likely: I purposefully stayed loose about the
| "infinite processes" to avoid going awry. I do however
| remembered him justifying his views as such though: he's
| not going into details, but he's making that point
| here[0] (c. 0:40). I assumed -- perhaps wrongfully --
| that he got those historical "facts" correct.
|
| https://youtu.be/I0JozyxM1M0?si=IFdWcEWNeNKDid7t&t=39
| kevindamm wrote:
| I found it helpful in some of my University math classes when
| I actually took the time to read the biographies that some of
| the textbooks included, the classes when I skimmed past them
| I did not remember details of the proof formulae for. But I
| don't know if this says more about the aid of history to the
| process of remembering or about my a priori interest in the
| topic for those particular classes.
|
| For a really good example of integrating the history along
| with the mathematics, and much more accessible than those
| math texts, I would recommend "Journey Through Genius" by
| Dunham[0]. It may be a little dated (published in 1990) and
| its focus is limited to algebra, geometry, number theory, and
| the history is perhaps too Western-biased, but it's good and
| it's short. Its material would make a solid foundation to
| build on top of because, in addition to the historical
| context, it shows a lot of the thought process into
| approaching certain landmark problems.
|
| [0]: https://www.goodreads.com/en/book/show/116185
| InitialLastName wrote:
| Boyer/Merzbach "A History of Mathematics" [0] is a tome in
| that vein. It spends a lot of time discussing the (mind-
| boggling, to a modern-mathematics-educated reader) methods
| that ancient peoples used to do real math (e.g. for
| engineering) as a way to motivate the development of the
| features of modern symbolic mathematics.
|
| [0] https://www.wiley.com/en-
| us/A+History+of+Mathematics%2C+3rd+...
| mamonster wrote:
| When I was an undergrad doing the mandatory measure theory
| course, I stumbled on a super old book(pre-1950 if I remember
| correctly, library card showed 3 people taking it out in the
| last 10 years) ,the name of which I forgot, that basically
| "re-built" the process that Lebesgue/Caratheodory/Riemann/etc
| followed, the problems they encountered (i.e Vitali set), why
| Lebesgue measure was the way it was and so on.
|
| I really wish I could remember the name of the book, but it
| made so much more sense than how even something like Stein
| Shakarchi or Billingsley, which introduced measures by either
| simply dumping the Vitali set on you as the main motivation
| or just not really explaining why stuff like outer
| measure/inner measure made sense.
| BeetleB wrote:
| > 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.
|
| Be careful.
|
| In undergrad, I worked as a math tutor at their tutoring
| institute. We had two calculus classes - one for
| engineers/science and the other for business/economics.
|
| If you're into Newtonian stuff or it's relevant to your degree,
| then your approach is all great. What I consistently saw is
| people in degrees like biology (or even industrial engineering)
| had a harder time learning _because_ of those physics
| applications in the book. They came in with one problem: Having
| trouble with the calculus. And then they discovered they had
| _two_ problems: To understand calculus they suddenly had to
| understand physics as well.
|
| You'd get students who were totally adept at differentiating
| and integrating, but would struggle with the problems that
| involved physics. Yes, it is important to be able to translate
| real world problems to math ones, but it's a bigger problem
| when you don't care about that particular field.
|
| Same problem with the business students. Since all the tutors
| were engineering folks, they had trouble tutoring the business
| students because their textbook was full of examples that
| required basic finance knowledge (and it really doesn't help
| that financial quantities are _discrete_ and not _continuous_ ,
| adding to the tutors' confusion).
|
| If you can find an application the student is interested in,
| then by all means, use that approach. For a general purpose
| textbook, though, it hinders learning for many students who
| don't care for the particular choice of application the book
| decided to use.
| dgacmu wrote:
| I got to see that yesterday with my daughter: her algebra
| book used PV=nRT as an example of joint proportion and asked
| some questions about it. She, having never seen the ideal gas
| law, was quite thrown by it.
|
| That said, after we went through it and had a brief physics
| lesson, it worked quite well and I'm glad they used the
| example instead of just making something up -- but it
| required having a tutor (me) on hand to help make the context
| make sense.
| HiroshiSan wrote:
| Ah yes I know that exact problem. AoPS intro to algebra.
| That question was great for getting an intuitive sense of
| proportion, though I do remember one of the sub problems
| gave me some trouble.
| dgacmu wrote:
| That's the one! It was a very nice example. I suspect for
| some students they could ignore the physics, but daughter
| needed to walk through the physical interpretation of the
| components before getting into the math.
|
| From my perspective as a tutor, it was a good use of time
| (gotta learn it some day anyway, and it provides useful
| physical intuition throughout life), but I could see it
| causing frustration if someone just wanted to learn
| algebra or didn't have a resource to turn to.
|
| (Love those books. I went and asked all of my colleagues
| who had won teaching awards, what books they recommended,
| and all of them said aops)
| acchow wrote:
| > required having a tutor (me) on hand to help make the
| context make sense.
|
| I wonder how good ChatGPT would be as that tutor. You can
| ask it to "explain like I'm 12"
| BeetleB wrote:
| Now imagine you were reading a mathematics textbook, and
| they use an example from something you really have no
| interest in (which for me would be finance). It includes
| terminology you have to learn - terminology that has no
| bearing on the math.
|
| As discouraging it is to learn math without context, it's
| even more discouraging to learn it in a context you hate.
| Koshkin wrote:
| I have no interest in finance but I liked this book:
|
| https://nononsense.gumroad.com/l/physicsfromfinance
| freestyle24147 wrote:
| If someone is having trouble understanding the absolute most
| basic part of classical mechanics (masses undergoing
| acceleration) something tells me they're not going to
| understand Calculus no matter what lens you view it through.
| It's not even about the equations, it's simply using it as a
| backstory for the motivation of the mathematics.
|
| > If you can find an application the student is interested
| in, then by all means, use that approach. For a general
| purpose textbook, though, it hinders learning for many
| students who don't care for the particular choice of
| application the book decided to use.
|
| This seems absurd. Just because some people will find a
| particular application less interesting, I don't think the
| answer is to throw out ALL applications and turn it into a
| generic boring slog through which no one will be able to see
| when and how it's useful.
| VyseofArcadia wrote:
| > It's not even about the equations, it's simply using it
| as a backstory for the motivation of the mathematics.
|
| Bingo. I rarely (if ever) used motivating examples from
| physics as test or homework problems. This was to ground
| calculus in reality somewhere. This was after years of
| wondering how to deal with the common student complaint of,
| "but why, where does this even come from?"
|
| So I started telling them where it comes from.
| calf wrote:
| The alternative is to ground it in philosophy and
| theoretical mathematics which would be even more
| abstruse...
| acchow wrote:
| Most people don't need or care to learn calculus. They just
| want to learn how to _use calculus_
| BeetleB wrote:
| > something tells me they're not going to understand
| Calculus no matter what lens you view it through.
|
| From my experience as a tutor, you are quite wrong in
| coming to that conclusion. Most people conflate mass and
| weight all the time, and have a fuzzy understanding of
| acceleration. It's not because they're thick in the head
| and incapable, but because _they don 't care_. That doesn't
| mean they don't care about other applications where math
| can help.
|
| And standard calculus textbooks go beyond what you are
| describing. All the ones I encountered would have the
| integral of force with displacement to get work (which
| confuses people when they hear it's the same as "energy").
|
| > This seems absurd.
|
| The truth often does seem so.
| chrisweekly wrote:
| Yeah!
|
| You're probably already aware of https://betterexplained.com --
| an amazing resource that exemplifies this same mindset.
| VyseofArcadia wrote:
| I was not, thank you for this.
| alejohausner wrote:
| When I took real analysis, I didn't get an intuitive sense of
| limits and all that delta/epsilon stuff. It was too abstract.
| Strangely enough, when I read David Foster Wallace's
| "Everything and More" on the history of infinity it all made
| more sense, because he described all the paradoxes and dead
| ends that mathematicians had run into before Cauchy and others
| brought rigor to the problem. Wallace was a postmodern
| novelist. I don't know why he described infinity so clearly; I
| didn't enjoy his other work.
| Koshkin wrote:
| Interesting... This e-d stuff is needed when you talk about
| functions, but when you start instead with the limits of
| numeric sequences, which are easy to grasp, the function
| limits come easy as well, because of the analogy between the
| ways these are defined.
| ForOldHack wrote:
| Chapter X is brilliant. Galois was very brilliant. He is always
| my #1 choice for ever of who, if anyone from history, I would
| like to have lunch with.
|
| He was introduced to me by my 5th grade calculus teacher Mr
| Steven Giavant, PhD. Galois bridged several disciplines and
| invented a new area of math.
|
| Most unfortunately he met his end extremely tragically.
|
| https://en.wikipedia.org/wiki/%C3%89variste_Galois
| XMPPwocky wrote:
| Absolutely agree. When I was a kid, I took a MOOC (MITx 6.002x)
| because I was interested in EE. They said calculus was a
| prerequisite, which I didn't know yet, but I went ahead
| anyways; every time the course required calculus, I'd go read
| the relevant chapters in Strang's _Calculus_. Felt incredibly
| natural, and I ended up going through the rest of Strang just
| for fun- probably the best learning experience I 've ever had,
| and I doubt I'd have ever done it without having the MOOC there
| to motivate the problem.
| gowld wrote:
| Notes, Videos, and Problems:
| https://www.maths.ed.ac.uk/~tl/galois/#notes
|
| Direct link to PDF of notes: https://arxiv.org/pdf/2408.07499
| mkw5053 wrote:
| 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 wrote:
| Also, it assumes little to no advanced math knowledge and can
| be found for free online :)
| sifar wrote:
| I second this book, really accessible. It taught me Abstract
| Algebra when I was learning it by myself.
| PreInternet01 wrote:
| [removed by author]
| ogogmad wrote:
| 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 wrote:
| 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 wrote:
| That is definitely not what Galois Fields are about.
| koolala wrote:
| 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 wrote:
| 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
|
| 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.
| koolala wrote:
| I can see how the idea extends naturally and also how it
| doesn't extend naturally. Thanks for explaining, I wonder
| what Galois would do if alive today with computers.
| graemep wrote:
| Patent threats can be used to stop something being taught,
| rather than being used? That is awful.
| koolala wrote:
| My memory of the story was they got some research funding
| from the troll company and were naive about human greed.
| koolala wrote:
| I had no idea Plank's work connected to Galois. I wish a
| University was able to publish free open information but money
| decides research.
| ogogmad wrote:
| Interestingly, there's a close connection between the
| "Fundamental Theorem of Galois Theory" and the "Fundamental
| Theorem of Covering Spaces".
| HPsquared wrote:
| 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
| gnulinux wrote:
| 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 wrote:
| 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 in a way that is
| hopefully accessible to more down-to-earth programmers and
| engineers.
| fredilo wrote:
| 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.
| klyrs wrote:
| Welcome to HN! FYI, you can edit comments for two hours
| after posting them.
| raldi wrote:
| Clicked around for a few minutes and couldn't find a sentence
| beginning, "Galois Theory is..."
| chii wrote:
| 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 ;so
| hoping you can get an understanding of it in a few sentences is
| probably asking too much tbh.
| skulk wrote:
| 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?
| taneliv wrote:
| 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_rD... (link copied from the page, I hope it
| works)
| Joker_vD wrote:
| > 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 wrote:
| Stack the vectors up so it's a matrix and take the determinant.
| The sign tells you which one it is.
| gowld wrote:
| 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 wrote:
| 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 wrote:
| Formulated differently, you cannot determine left- and
| right-handedness but you can determine same-handedness.
| gowld wrote:
| 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.)
| kevinventullo wrote:
| Once you get to higher extensions where the roots are
| indistinguishable w.r.t. the base field, mathematicians will
| just write something like
|
| "Let a_1, ..., a_5 be the five roots of p(x)."
| fredilo wrote:
| The sentence you quote actually gives you the answer to your
| question but it is not completely obvious why. :)
|
| The complex numbers are essentially the theory of 2D-space. You
| are asking about 3D-space. The statement that you quoted tells
| you that you cannot distinguish between up and down in theory.
|
| Now, 2D-space is part of every 3D-space. There are multiple
| ways to see this. The easiest is to just drop the z-coordinate.
|
| Suppose you could distinguish left-handed and right-handed in
| 3D-space. In this case, you would have a way to distinguish up
| and down in the embedded 2D-space. However, you cannot do this
| distinction in 2D-space and therefore you cannot distinguish
| left-handed and right-handed in 3D-space.
| revskill wrote:
| 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 wrote:
| 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 wrote:
| I think everyone already knows some math before they start
| reading books on it.
| senderista wrote:
| Ian Stewart's book is excellent for self-study and has some
| fascinating historical background.
|
| https://www.taylorfrancis.com/books/mono/10.1201/97810032139...
| susam wrote:
| Galois Theory by Ian Stewart is an excellent book indeed! I've
| got a hard copy lying at home that I am currently reading
| slowly page by page. I am planning to host book club meetups
| with this book later this year, perhaps during the winter if I
| am able to make good progress with this book.
|
| In the meantime, if there is someone here who is interested in
| reading this type of books together and share updates with each
| other, I'd like to invite you to the IRC and Matrix channel
| named #bitwise [1][2] (the IRC and Matrix channels are bridged
| together, so you could join either one of them). The channel
| consists of some HN users as well as some users from other
| channels like ##math, ##physics, #cs, etc. It serves as an
| online space to share updates about mathematics and computation
| books you are reading and discuss their content.
|
| [1] https://web.libera.chat/#bitwise
|
| [2] https://app.element.io/#/room/#bitwise:matrix.org
| nyankosensei wrote:
| Another introduction from an historical point of view is
| "Galois Theory for Beginners: A Historical Perspective" by Jorg
| Bewersdorff
|
| https://bookstore.ams.org/view?ProductCode=STML/95
| jorgenveisdal wrote:
| Love this!
| fredgrott wrote:
| Do not forget the numbers book covering history of numbers that
| Albert Einstein recommended....author is Tobias Dantzig...
| amai wrote:
| I guess you mean this one:
| https://en.wikipedia.org/wiki/Number:_The_Language_of_Scienc...
| enugu wrote:
| 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/
| artemonster wrote:
| I wonder, can you do an LLM in GF(2)?
| klyrs wrote:
| Somebody did a paper on GF(3)...
|
| https://arxiv.org/abs/2402.17764
| zengid wrote:
| ELI5 what Galois theory is?
| klyrs wrote:
| 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 wrote:
| how hard is it to say "an advanced theory about algebra",
| instead of downvoting?
| daitangio wrote:
| Btw, the life of Galois is quite interesting: he died very young,
| and was a quite clever mathematician...
| klyrs wrote:
| > ... 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 wrote:
| Naming the groups of order 8 is harder than naming all the
| smaller groups.
| marshallward wrote:
| > I hope you can list all of the groups of order < 8 without
| having to think too hard.
|
| Welp, guess I'm out.
| Koshkin wrote:
| https://en.wikipedia.org/wiki/List_of_small_groups
| Koshkin wrote:
| _Galois Theory For Beginners_ by John Stillwell is the shortest
| introduction that I 've ever seen.
|
| https://www.scribd.com/document/81010821/GaloisTheoryForBegi...
| dmd wrote:
| https://chalkdustmagazine.com/blog/review-of-galois-knot-the...
| is probably the best review.
| gowld wrote:
| This is a joke not related to the submission or its topic.
| broabprobe wrote:
| 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/
| Venkatesh10 wrote:
| The website is just plethora of knowledge and content in 90s
| design. Just pure bliss and I love it.
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