Source of Name
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.