Abel-Ruffini Theorem

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Theorem

There is no general algebraic solution for determining all the roots of a polynomial of degree $5$ or higher.


Proof


This theorem requires a proof.
In particular: Galois theory seems the best choice; we probably also want the actual proof by Abel
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The general polynomial:

$\map f x = x^n - s_1 x^{n - 1} \cdots \paren {-1}^n s_n$

has Galois group $S_n$.

From Symmetric Group on Greater than 4 Letters is Not Solvable:

$S_n$ is not solvable for $n \ge 5$.

By Polynomial is Solvable by Radicals iff Galois Group is Solvable

the general polynomial is not solvable by radicals.

$\blacksquare$


Source of Name

This entry was named for Niels Henrik Abel and Paolo Ruffini.


Historical Note

The Abel-Ruffini Theorem, on the general insolubility of the quintic by radicals, was stated by Paolo Ruffini, who built an incomplete proof in $1799$.

This was published in his two-volume work La teoria generale delle equazioni.


Niels Henrik Abel provided the first complete proof in $1823$.

He published this in a small pamphlet Mémoire sur les équations algébriques ou on démontre l'impossibilité de la résolution de l'équation générale du cinquième degré ($1824$) at his own expense.

He sent a copy of this to Carl Friedrich Gauss, but for some reason Gauss put it aside and never opened it.

Thus Abel never had cause to visit Gauss and the pair never met.


It later transpired that Évariste Galois had independently proved this theorem some years earlier, in a work that was not published in $1846$, some $25$ years after his death.

Moreover, Galois' analysis of the problem also gave a complete answer to the question of which equations are solvable in radicals and which are not.


Sources

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