Let f(x) be an irreducible polynomial of odd degree n > 1 whose Galois group is a Frobenius group. We suppose that the Frobenius complement is a cyclic group of even order h. Let 2t h. For each i = 1, 2,..., t we show that the splitting field L of f(x) has exactly one subfield Ki with [Ki : ℚ] = 2i. These subfields form a tower of normal extensions ℚ ⊂ K1 ⊂ K2 ⊂ .... ⊂ Kt with [Ki : Ki-1] = 2 (i = 1, 2,..,t) and K0 = ℚ. Our main result in this paper is an explicit formula for an element αi in Ki-1 such that K i = ℚ(√αi) (i = 1, 2,..., t). This result is applied to DeMoivre's quintic x5 - 5ax3 + 5a2x - b, solvable quintic trinomials x5 + ax + b, as well as to some numerical polynomials of degrees 5, 9, and 13.

Additional Metadata
Keywords Frobenius group, Galois group, Subfields of splitting field
Journal Communications in Algebra
Spearman, B.K. (Blair K.), Williams, K.S, & Yang, Q. (Qiduan). (2003). The 2-Power Degree Subfields of the Splitting Fields of Polynomials with Frobenius Galois Groups. Communications in Algebra, 31(10), 4745–4763.