On the Lagrange resolvents of a dihedral quintic polynomial (Q1777897)

The cyclic quartic field generated by the fifth powers of the Lagrange resolvents of a dihedral quintic polynomial \(f(x)\) is explicitly determined in terms of a generator for the quadratic subfield of the splitting field of \(f(x)\). If \(r_1,r_2,r_3\) and \(r_4\) denote the Lagrange resolvents of a fixed root of \(f\) and set \(R_i=r^5_i\) for \(i=1,2,3,4\) then it is proved that \(\mathbb Q(R_i)=\mathbb Q(\sqrt{-m(5+2\sqrt 5)})\) is a cyclic quartic extension of \(\mathbb Q\). Reference is made to \textit{D. S. Dummit} [Math. Comput. 57, 387--401 (1991; Zbl 0729.12008)].