Why no general solution to quintic equations?

It seems odd to me that we have no general solution to quintic equations yet.
Is it possible that it exist any general solutions to equations of the fifth degree (and higher) that just haven't been discovered yet?
Or are we certain that it doesn't exist?

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Office_Shredder
Staff Emeritus
Gold Member
It is in fact a theorem that no solution exists that only uses functions that we would consider typical.

http://en.wikipedia.org/wiki/Abel–Ruffini_theorem

The theorem says that you cannot solve the general fifth degree polynomial using only basic arithmetic operations and calculating nth roots of numbers. It might be that there are other functions which you can allow yourself to use to solve higher degree polynomials (a trivial example is that if you define a function P(a,b,c,d,e) to return a root for a polynomial whose coefficients are a,b,c,d and e then you have "solved" the polynomial).

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Thanks. I don't understand why it don't exist yet and I don't understand the proof completely. Just out of curiosity, how much math do you need to read to understand the proof?

Thanks. I don't understand why it don't exist yet
It exists yet: The roots of the quintic equation can be analytically expressed thanks to the Jacobi theta functions, but not with a finie number of elementary functions.

jbunniii
Homework Helper
Gold Member
Thanks. I don't understand why it don't exist yet and I don't understand the proof completely. Just out of curiosity, how much math do you need to read to understand the proof?
A fairly large portion of a typical undergraduate course in abstract algebra. For example, the proof using Galois theory is given in the final chapter (33) of Pinter's A Book of Abstract Algebra, and it depends to some degree on almost all of the first 32 chapters.

There are more direct proofs that do not use Galois theory - indeed, Abel's original proof predated Galois theory. Here is a video which sketches one such proof:

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SteamKing
Staff Emeritus