The relationship between solvability of ruler and compass problems and solvability of algebraic equations by radicals

Question

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.)

Answer 1

The idea of coding ruler and compass constructions algebraically was more or less known to Vieta, as described in Supplementum Geometriae (1593), see Viète's Relevance and his Connection to Euler. It was developed further by Descartes in La Geometrie (1637), where the idea of constructing segment arithmetic (including square roots) geometrically also appears. Gauss was first to investigate the algebraic/geometric correspondence (what we would call) rigorously, and proved that certain regular polygons can be inscribed into the circle by converting the geometric problem into one about the cyclotomic fields (1796). He claimed that only those he described are constructible, but there is no proof in his Nachlass.

The proof was only given by Wantzel in 1837 (along with a proof of the impossibility of trisection), 5 years after Galois' death, although he did not rely on his work, which remained little known for a while. Explicitly, in modern terms, the correspondence between straightedge and compass constructions and quadratic extensions is proved and utilized extensively in Hilbert's Grundlagen der Geometrie (1899). It is based on developing Descartes's segment arithmetic.

Answer 2

There is no "equivalence of ruler and compass solvability and solution by radicals". For example, cubic root of $2$ is evidently a radical but cannot be constructed with a compass and ruler. The relation between construction and solvability in radicals is the following: only those segments can be constructed whose length is expressed by arithmetic operations and extraction of SQUARE roots. The result is due to Gauss, and it pre-dates Galois theory. It was Gauss who determined completely which regular $n$-gons can be constructed with a compass and a ruler.

Ref. Robin Hartshorne, Geometry: Euclid and beyond, Springer-Verlag, New York, 2000.