Solvability of cubic and quartic equations using one radical

Abstract

Theorem. An irreducible cubic polynomial with rational coefficients has a root in a one step radical extension of Q if and only if the discriminate is a square of a rational number. Theorem. An irreducible polynomial x^4+px^2+qx+s with rational coefficients q\ne0, p and s has a root in a one step radical extension of Q if and only if the cubic resolution has rational root t such that t>p/2 and A:=16(t^2-s)^2-(t^2-s)(2t+p)^2 is a square of a rational number.