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

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.)

• I'm virtually certain an answer can be found in many of the references on pp. 36-41 of this manuscript, but I don't have time now to look into this. Note that many of those references have URLs to digitized versions that are freely available online. – Dave L Renfro Jun 16 at 23:04
• @DaveLRenfro Thanks for your reference. Luckily my question has been answered here. Interesting paper btw. – Nick R Jun 17 at 16:35

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.