# Formula for roots of polynomials

For a quadratic polynomial there exists a formula for its roots. I read that similarly for polynomials of degree 3 and 4 there also exists such a formula but that no such formulas exist for polynomials of degree 5 or higher.

Does there exist a proof that no such formulas for $\ge 5$ can exist or is it that they have not been found yet?

• Yes, such a proof exists, as answered by Jared. It was proven by the Norwegian mathematician Niels Henrik Abel (1802-1829). Aug 12, 2013 at 8:13
• Actually, formulas do exist for some- but-not-all polynomials of degree 5-or-higher. This depends on the Galois group of the polynomial. This is part of Galois Theory; you calculate the Galois group and decide if it is solvable, I believe.
– FBD
Aug 12, 2013 at 8:13
• Am I correct in thinking that, even though no general formula exists (i.e. for all possible solutions), this does not mean to say that the solutions themselves do not exist? Aug 12, 2013 at 8:16
• @pbs, all this means there is no general solution by means of radicals (say, as in the quadratic, cubic or quartic case) of a quintic equation. Aug 12, 2013 at 8:43
• See the book Abel's Proof by Peter Pesic.
– lhf
Aug 15, 2013 at 19:24

## 2 Answers

There is a proof, and it is quite beautiful. The result is known as the Abel-Ruffini Theorem, and the standard proof of this fact uses Galois theory and the fact that the alternating group on $n\ge 5$ symbols is not solvable. This wasn't the original proof, but it is the most elegant. The link provided contains a brief outline of the proof based on Galois's ideas.

There does exists a proof that polynomials with degree greater than 4 cannot be solved with radicals, the proof relies heavily on Galois theory. See for example here.