# Abel-Ruffini Theorem/Historical Note

## Historical Note on Abel-Ruffini Theorem

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

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Introduction - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $5$ - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.27$: Abel ($\text {1802}$ – $\text {1829}$) - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $5$