The subfields of the splitting field of a solvable quintic trinomial

Let Q denote the field of rational numbers, and set Q * = Q\{O). Let a c Q* and b c Q * be such that the quintic trinomial, f (X) = X6 +aX+b is both irreducible and solvable. Polynomials of this type are characterized in [3, Theorem]. Let L denote the splitting field off. Ltt r denote the unique retional root of the resolvent sextic of XS +aX+b [3, eqn. (17)l; and set so that c is a nonnegative rational numbcr, = f 1, and e is a nonzero rational number. I t is shown in [3] that The Galois group Gt of f is the dihedral group D, of order 10 if 5(c2 + 1) 4 2, and is the Frobenius group F,, of order 20 if 5(02.+ 1) B Q2. If Gt = D, then Gt has five subgroups of order 2, and one of order 5. The five quintic subfields of L are Q(R),i =;..lt2,3,4,5, wher