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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?

- Thread starter Superhoben
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- #1

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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?

- #2

Office_Shredder

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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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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.Thanks. I don't understand why it don't exist yet

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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

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

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