Solving solvable quintics
David S. Dummit · Mathematics of Computation · 1991
Let $f(x) = {x^5} + p{x^3} + q{x^2} + rx + s$ be an irreducible polynomial of degree 5 with rational coefficients. An explicit resolvent sextic is constructed which has a rational root if and only if $f(x)$ is solvable by radicals (i.e., when its Galois group is contained in the Frobenius group ${F_{20}}$ of order 20 in the symmetric group ${S_5}$). When $f(x)$ is solvable by radicals, formulas for the roots are given in terms of p, q, r, s which produce the roots in a cyclic order.