
Why is the Quintic Unsolvable? - luu
https://www.akalin.com/quintic-unsolvability
======
jaymzcampbell
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-
Mathem...](https://www.amazon.co.uk/Equation-That-Couldnt-Solved-
Mathematical/dp/0743258215)

[2] [https://www.amazon.co.uk/Fearless-Symmetry-Exposing-
Patterns...](https://www.amazon.co.uk/Fearless-Symmetry-Exposing-Patterns-
Numbers/dp/0691138710)

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acidburnNSA
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/-H...](https://groups.google.com/forum/#!topic/sci.math.symbolic/-HhMjcfbW3k)

~~~
wfunction
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...

~~~
avian
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)

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ice109
just today cleaning out my desk i found this again

[http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwel...](http://www.math.jhu.edu/~smahanta/Teaching/Spring10/Stillwell.pdf)

>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.

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ocfnash
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
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
No that wouldn't work. Consider the equation x^5 + 1 = 0. Clearly that isn't
solvable by x=0.

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jacobolus
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
As per
[https://news.ycombinator.com/item?id=14685892](https://news.ycombinator.com/item?id=14685892)
this is also by Arnold.

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

~~~
enriquto
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
All quintics have solutions, but there is no general quintic formula.

~~~
moomin
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
What other operation? If we don't know what it is, how do you know it needs
just one?

~~~
vog
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+a1 _x+...+a4_ x^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.)

~~~
moomin
I think it's something like p(x) = smallest positive solution of z^5 + z - x.
Compare sqrt(x) = smallest positive solution of z^2 - x.

But it's a _long_ time since I've done Galois theory and I can't find a decent
math exchange answer for it right now, so don't treat this as gospel.

~~~
akalin
Yeah, I think you're thinking of a Bring radical:
[https://en.wikipedia.org/wiki/Bring_radical](https://en.wikipedia.org/wiki/Bring_radical)

Edit: Oh, but that only lets you solve some quintics.
[https://news.ycombinator.com/item?id=14686886](https://news.ycombinator.com/item?id=14686886)
describes the functions you need to solve all quintics.

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acjohnson55
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_const...](https://en.m.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction)

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knlje
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
Nobody was saying otherwise.

~~~
zucchini_head
The parent comment wasn't saying that nobody was saying otherwise...

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

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dingo_bat
Link to cached version since original is down:
[http://cc.bingj.com/cache.aspx?q=why+is+the+quintic+unsolvab...](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)

~~~
ktta
Wow. People still use Bing?

~~~
popcorncolonel
Bing rewards baby

~~~
Walf
"Good baby; have some more SERPs."

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

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

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mrcactu5
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.

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crb002
Modulo 5 it is the full transformation semigroup on 5 elements.

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tobbe2064
Very nice, Thank you Always wanted to know, now you saved me a year dedicated
to abstract algebra

