8 ms·

Why is the Quintic Unsolvable?

216 points by luu 9y ago

ice109 9y ago
just today cleaning out my desk i found this again http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwell.pdf 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.
- deleted 9y ago
  [deleted]
acidburnNSA 9y ago
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]! [1] https://store.wolfram.com/view/misc/popup/solving-tqp.html https://store.wolfram.com/view/misc/popup/solving-tqp.html [2] https://groups.google.com/forum/#!topic/sci.math.symbolic/-HhMjcfbW3k https://groups.google.com/forum/#!topic/sci.math.symbolic/-H...
- wfunction 9y ago
  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...
  - wiredfool 9y ago
    They bought DejaNews, who grabbed a bunch of old net.news archives from various places. They then shoehorned it into the Groups interface.
  - avian 9y ago
    Not mailing lists. These are Usenet posts. Years ago Google bought a service that was archiving the Usenet network and integrated their data into Google Groups. https://en.wikipedia.org/wiki/Usenet https://en.wikipedia.org/wiki/Usenet
- SonOfLilit 9y ago
  Ordering the poster to Israel: poster $10, shipping $52. Wat.
jacobolus 9y ago
Folks might enjoy the lectures about this which V.I. Arnold taught to 14–16 year old students, https://www.mathcamp.org/2015/abel/abel.pdf https://www.mathcamp.org/2015/abel/abel.pdf
- chx 9y ago
  As per https://news.ycombinator.com/item?id=14685892 https://news.ycombinator.com/item?id=14685892 this is also by Arnold.
- chii 9y ago
  wow, each increase in degree of the equation is literally an order of magnitude more complicated than before!
- mrcactu5 9y ago
  I have a master's in Math. none of it is new, but I'm so glad someone wrote this. expanding out all the key points.
cft 9y ago
Interestingly it was first proved by Abel when he was 22 years old.
dingo_bat 9y ago
Link to cached version since original is down: http://cc.bingj.com/cache.aspx?q=why+is+the+quintic+unsolvable%3f+-+fred+akalin&d=4582117628184131&mkt=en-US&setlang=en-US&w=V_hVzGqYXfAjtLGgeL-vBrmM_pi8QPUN http://cc.bingj.com/cache.aspx?q=why+is+the+quintic+unsolvab...
- ktta 9y ago
  Wow. People still use Bing?
  - popcorncolonel 9y ago
    Bing rewards baby
    - Walf 9y ago
      "Good baby; have some more SERPs."
  - dingo_bat 9y ago
    I recently (couple months ago) switched to bing and edge, I am fed up of google's abuse of power. I must say I don't find myself constrained.
crb002 9y ago
Modulo 5 it is the full transformation semigroup on 5 elements.
akalin 9y ago
Author here, in case anyone has any questions. :)
- enriquto 9y ago
  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.
  - dullcrisp 9y ago
    All quintics have solutions, but there is no general quintic formula.
    - enriquto 9y ago
      Of course, but in this game "solution" means "a number that can be expressed as a composition of radicals and elementary operations".
    - moomin 9y ago
      Not true, there's only no general quintic formula that employs a specific, restricted set of operations. In practice, there is a general quintic formula. It just needs one extra operation.
      - popcorncolonel 9y ago
        What other operation? If we don't know what it is, how do you know it needs just one?
        vog 9y ago
        Does that matter? If you are free to add any additional operation, this whole thing becomes meaningless. You can simply define your operation "NewOp(a0,a1,..,a4)" to something like "the smallest root of the quintic a0+a1x+...+a4x^4+x^5". (Here, "smallest" can be anything as long as it is a completely defined tie-breaker, such as: the value with the smallest real part, and among those the one with the smallest imaginary part.)
tobbe2064 9y ago
Very nice, Thank you Always wanted to know, now you saved me a year dedicated to abstract algebra
fspeech 9y ago
Very nice! The covering transforms have similar structure to the Galois theory. It is cool to see them in action visually.
ocfnash 9y ago
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.
- lightedman 9y ago
  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.
  - openasocket 9y ago
    No that wouldn't work. Consider the equation x^5 + 1 = 0. Clearly that isn't solvable by x=0.
  - jacksnipe 9y ago
    That's not a "solution" in the algebraic sense that we're talking about.
knlje 9y ago
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.
- IshKebab 9y ago
  Nobody was saying otherwise.
  - zucchini_head 9y ago
    The parent comment wasn't saying that nobody was saying otherwise...
jaymzcampbell 9y ago
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. [1] https://www.amazon.co.uk/Equation-That-Couldnt-Solved-Mathematical/dp/0743258215 https://www.amazon.co.uk/Equation-That-Couldnt-Solved-Mathem... [2] https://www.amazon.co.uk/Fearless-Symmetry-Exposing-Patterns-Numbers/dp/0691138710 https://www.amazon.co.uk/Fearless-Symmetry-Exposing-Patterns...
nimish 9y ago
If you extend the operations allowed it's possible using some exotic functions: https://en.wikipedia.org/wiki/Thomae%27s_formula#cite_note-thomae-3 https://en.wikipedia.org/wiki/Thomae%27s_formula#cite_note-t...
mrcactu5 9y ago
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.
acjohnson55 9y ago
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? https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Dickson_const...

216 points by luu 9y ago

just today cleaning out my desk i found this again http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwell.pdf 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.

deleted 9y ago
[deleted]

acidburnNSA 9y ago

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]! [1] https://store.wolfram.com/view/misc/popup/solving-tqp.html https://store.wolfram.com/view/misc/popup/solving-tqp.html [2] https://groups.google.com/forum/#!topic/sci.math.symbolic/-HhMjcfbW3k https://groups.google.com/forum/#!topic/sci.math.symbolic/-H...

wfunction 9y ago
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...
- wiredfool 9y ago
  They bought DejaNews, who grabbed a bunch of old net.news archives from various places. They then shoehorned it into the Groups interface.
- avian 9y ago
  Not mailing lists. These are Usenet posts. Years ago Google bought a service that was archiving the Usenet network and integrated their data into Google Groups. https://en.wikipedia.org/wiki/Usenet https://en.wikipedia.org/wiki/Usenet
SonOfLilit 9y ago
Ordering the poster to Israel: poster $10, shipping $52. Wat.

jacobolus 9y ago

Folks might enjoy the lectures about this which V.I. Arnold taught to 14–16 year old students, https://www.mathcamp.org/2015/abel/abel.pdf https://www.mathcamp.org/2015/abel/abel.pdf

chx 9y ago
As per https://news.ycombinator.com/item?id=14685892 https://news.ycombinator.com/item?id=14685892 this is also by Arnold.
chii 9y ago
wow, each increase in degree of the equation is literally an order of magnitude more complicated than before!
mrcactu5 9y ago
I have a master's in Math. none of it is new, but I'm so glad someone wrote this. expanding out all the key points.

cft 9y ago

Interestingly it was first proved by Abel when he was 22 years old.

dingo_bat 9y ago

Link to cached version since original is down: http://cc.bingj.com/cache.aspx?q=why+is+the+quintic+unsolvable%3f+-+fred+akalin&d=4582117628184131&mkt=en-US&setlang=en-US&w=V_hVzGqYXfAjtLGgeL-vBrmM_pi8QPUN http://cc.bingj.com/cache.aspx?q=why+is+the+quintic+unsolvab...

ktta 9y ago
Wow. People still use Bing?
- popcorncolonel 9y ago
  Bing rewards baby
  - Walf 9y ago
    "Good baby; have some more SERPs."
- dingo_bat 9y ago
  I recently (couple months ago) switched to bing and edge, I am fed up of google's abuse of power. I must say I don't find myself constrained.

crb002 9y ago

Modulo 5 it is the full transformation semigroup on 5 elements.

akalin 9y ago

Author here, in case anyone has any questions. :)

enriquto 9y ago
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.
- dullcrisp 9y ago
  All quintics have solutions, but there is no general quintic formula.
  - enriquto 9y ago
    Of course, but in this game "solution" means "a number that can be expressed as a composition of radicals and elementary operations".
  - moomin 9y ago
    Not true, there's only no general quintic formula that employs a specific, restricted set of operations. In practice, there is a general quintic formula. It just needs one extra operation.
    - popcorncolonel 9y ago
      What other operation? If we don't know what it is, how do you know it needs just one?
      - vog 9y ago
        Does that matter? If you are free to add any additional operation, this whole thing becomes meaningless. You can simply define your operation "NewOp(a0,a1,..,a4)" to something like "the smallest root of the quintic a0+a1x+...+a4x^4+x^5". (Here, "smallest" can be anything as long as it is a completely defined tie-breaker, such as: the value with the smallest real part, and among those the one with the smallest imaginary part.)

tobbe2064 9y ago

Very nice, Thank you Always wanted to know, now you saved me a year dedicated to abstract algebra

fspeech 9y ago

Very nice! The covering transforms have similar structure to the Galois theory. It is cool to see them in action visually.

ocfnash 9y ago

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.

lightedman 9y ago
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.
- openasocket 9y ago
  No that wouldn't work. Consider the equation x^5 + 1 = 0. Clearly that isn't solvable by x=0.
- jacksnipe 9y ago
  That's not a "solution" in the algebraic sense that we're talking about.

knlje 9y ago

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.

IshKebab 9y ago
Nobody was saying otherwise.
- zucchini_head 9y ago
  The parent comment wasn't saying that nobody was saying otherwise...

jaymzcampbell 9y ago

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. [1] https://www.amazon.co.uk/Equation-That-Couldnt-Solved-Mathematical/dp/0743258215 https://www.amazon.co.uk/Equation-That-Couldnt-Solved-Mathem... [2] https://www.amazon.co.uk/Fearless-Symmetry-Exposing-Patterns-Numbers/dp/0691138710 https://www.amazon.co.uk/Fearless-Symmetry-Exposing-Patterns...

nimish 9y ago

If you extend the operations allowed it's possible using some exotic functions: https://en.wikipedia.org/wiki/Thomae%27s_formula#cite_note-thomae-3 https://en.wikipedia.org/wiki/Thomae%27s_formula#cite_note-t...

mrcactu5 9y ago

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.

acjohnson55 9y ago

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? https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Dickson_const...