The 2-Power Degree Subfields of the Splitting Fields of Polynomials with Frobenius Galois Groups
Blair Kenneth Spearman, Kenneth S. Williams, Qiduan Yang · Communications in Algebra · 2003
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 2 t h. For each i = 1, 2,…, t we show that the splitting field L of f(x) has exactly one subfield K i with [K i : ℚ] = 2 i . These subfields form a tower of normal extensions ℚ ⊂ K 1 ⊂ K 2 ⊂ ċċċ ⊂ K t with [K i : K i−1] = 2 (i = 1, 2,…, t) and K 0 = ℚ. Our main result in this paper is an explicit formula for an element α i in K i−1 such that (i = 1, 2,…, t). This result is applied to DeMoivre's quintic x 5 − 5ax 3 + 5a 2 x − b, solvable quintic trinomials x 5 + ax + b, as well as to some numerical polynomials of degrees 5, 9, and 13.