Galois obtained necessary and sufficient conditions for an algebraic equation (in one variable) to be solvable by extraction of a chain of square roots. A beautiful application of this is to the problem of solvability of geometric problems by ruler and compass.
My question is, when was the equivalence of ruler and compass solvability and solution by radicals originally identified? Also, who first applied Galois’ results to prove that a regular polygon with a prime number $p$ of sides is constructible by ruler and compass if and only if $p$ is of the form $2^{2n}+1$. (We know that Gauss proved the “if” part, and that the same methods can be used to prove the impossibility of trisecting an angle and doubling a cube.)