The 2-Power Degree Subfields of the Splitting Fields of Polynomials with Frobenius Galois Groups
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.
|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.