It's really nice to see this on here! Around 8 years ago I read "The equation that couldn't be solved"[1] which is a very readable and not-mathy book about this topic. I was absolultely fascinated and ended up enrolling on an Open University mathematics degree which I've just completed. If you are interested in this and fancy something light to read this book is a nice distraction.
It was this problem and it's solution that truely opened my eyes up to the enormous power and abstraction of mathematics.
A much, much less readable book about Galois theory (this is really the cornerstone of the (general) quintic being unsolvable by a formula with radicals) is Fearless Symmettry [2]. That is a book I wish was twice the length, it will explain what a matrix is over pages but then do a drive by with Frobenius numbers. It is also let down by extremely poor typesetting on Kindle. However if you can stomach it, it's probably the only "popular" book on Galois theory that I know of. It focuses on Wiles proof of Fermats Last Theorem.
Around 1994 my dad bought this huge "Solving the Quintic" poster from Wolfram Research after he had started playing around with Mathematica. Staring at it grew a pretty good fascination in mathematics for me.
I just looked into it and it turns out you can still buy it [1]! Oh, and here's a 1994 announcement about it on sci.math.symbolic [2]!
Off-topic, but can someone explain to me how mailing lists and threads from 1994 end up on Google Groups? I never really understood what goes on there...
>The aim of this paper is to prove the unsolvability by radicals of the quintic (in fact of the general nth degree equation for n >= 5) using just the fundamentals of
groups, rings and fields from a standard first course in algebra.
essentially all of the fundamentals of galois theory in 5 short (very readable!) pages.
Another lovely aspect to this problem is that the quintic _is_ solvable if you allow yourself to use certain special functions naturally associated to the icosahedron.
The genesis of the relationship is that the group of rotations of the icosahedron is the same group, A_5.
It's solvable in many other ways. There's no rule in algebra that states each individual variable must be a different number. Thus, every variable = 0 then the equation is solved.
> Now this formula looks right, since x_1x and x_2x are at the same coordinates as r_1r and r_2r.
I assume you mean "since before they have been moved...", but by the time the reader gets to that point you've already told them to move r1 and r2 and x1 and x2 obviously aren't at the same coordinates.
Well, some quintics are solvable, aren't they? (like x^5-1=0). Maybe for each particular quintic there is a different formula that solves it. You just prove that there's no formula that works in all cases when you plug the coefficients at the same place.
It would be more satisfying (and more general) to exhibit a specific unsolvable quintic.
Thanks for this, this is a topic I've been curious about for a long time, but never knew how to learn about it without diving into a textbook.
I'm reading through it now, and I'm confused by this statement: "Second, and more surprisingly, if you swap r_1 and r_2r, x_1 and x_2 also exchange places, seemingly contradicting Theorem 1!"
Why is there any connection to theorem 1 here? We're talking about a radical, and theorem 1 is concerned with rational expressions.
Is there an analogy with the way the sequence from reals to complex numbers to quaternions to octonions (iterations of the Cayley-Dickson construction) loses algebraic properties?
Numerical methods quite easily give the roots of arbitrary polynomials of much higher order. QR-iteration works well up to some polynomial order, say at least 20. The idea is to construct a matrix that has the studied polynomial as its characteristic polynomial, and find the eigenvalues using a repeated QR decomposition. This gives you all complex and real roots.
he does a pretty good job of connecting unsolvability by radicals to the unsolvability of the permutation group S5.
I suspect what happens these quintics define Riemann surfaces and perhaps the premutation of the roots an be mapped unto the fundamental group of the surface.
[+] [-] jaymzcampbell|8 years ago|reply
It was this problem and it's solution that truely opened my eyes up to the enormous power and abstraction of mathematics.
A much, much less readable book about Galois theory (this is really the cornerstone of the (general) quintic being unsolvable by a formula with radicals) is Fearless Symmettry [2]. That is a book I wish was twice the length, it will explain what a matrix is over pages but then do a drive by with Frobenius numbers. It is also let down by extremely poor typesetting on Kindle. However if you can stomach it, it's probably the only "popular" book on Galois theory that I know of. It focuses on Wiles proof of Fermats Last Theorem.
[1] https://www.amazon.co.uk/Equation-That-Couldnt-Solved-Mathem...
[2] https://www.amazon.co.uk/Fearless-Symmetry-Exposing-Patterns...
[+] [-] acidburnNSA|8 years ago|reply
I just looked into it and it turns out you can still buy it [1]! Oh, and here's a 1994 announcement about it on sci.math.symbolic [2]!
[1] https://store.wolfram.com/view/misc/popup/solving-tqp.html
[2] https://groups.google.com/forum/#!topic/sci.math.symbolic/-H...
[+] [-] wfunction|8 years ago|reply
[+] [-] SonOfLilit|8 years ago|reply
[+] [-] ice109|8 years ago|reply
http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwel...
>The aim of this paper is to prove the unsolvability by radicals of the quintic (in fact of the general nth degree equation for n >= 5) using just the fundamentals of groups, rings and fields from a standard first course in algebra.
essentially all of the fundamentals of galois theory in 5 short (very readable!) pages.
[+] [-] unknown|8 years ago|reply
[deleted]
[+] [-] ocfnash|8 years ago|reply
The genesis of the relationship is that the group of rotations of the icosahedron is the same group, A_5.
[+] [-] lightedman|8 years ago|reply
[+] [-] jacobolus|8 years ago|reply
[+] [-] chx|8 years ago|reply
[+] [-] mrcactu5|8 years ago|reply
[+] [-] chii|8 years ago|reply
[+] [-] akalin|8 years ago|reply
[+] [-] IshKebab|8 years ago|reply
> Now this formula looks right, since x_1x and x_2x are at the same coordinates as r_1r and r_2r.
I assume you mean "since before they have been moved...", but by the time the reader gets to that point you've already told them to move r1 and r2 and x1 and x2 obviously aren't at the same coordinates.
[+] [-] alecbenzer|8 years ago|reply
> But that means that our candidate solution cannot be the quadratic formula! If it were, then x_1 and x_2 would have ended up swapped, too.
Why?
[+] [-] enriquto|8 years ago|reply
It would be more satisfying (and more general) to exhibit a specific unsolvable quintic.
[+] [-] 19eightyfour|8 years ago|reply
[+] [-] alanbernstein|8 years ago|reply
I'm reading through it now, and I'm confused by this statement: "Second, and more surprisingly, if you swap r_1 and r_2r, x_1 and x_2 also exchange places, seemingly contradicting Theorem 1!"
Why is there any connection to theorem 1 here? We're talking about a radical, and theorem 1 is concerned with rational expressions.
[+] [-] TheRealPomax|8 years ago|reply
[+] [-] HurrdurrHodor|8 years ago|reply
[+] [-] acjohnson55|8 years ago|reply
https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Dickson_const...
[+] [-] knlje|8 years ago|reply
[+] [-] IshKebab|8 years ago|reply
[+] [-] cft|8 years ago|reply
[+] [-] dingo_bat|8 years ago|reply
[+] [-] ktta|8 years ago|reply
[+] [-] fspeech|8 years ago|reply
[+] [-] nimish|8 years ago|reply
[+] [-] mrcactu5|8 years ago|reply
I suspect what happens these quintics define Riemann surfaces and perhaps the premutation of the roots an be mapped unto the fundamental group of the surface.
[+] [-] crb002|8 years ago|reply
[+] [-] tobbe2064|8 years ago|reply