Cyclotomic-intermediate Fields via Gauss Sums

Let p be an odd prime, and let m divide p−1. Let ζ = e and let ω = e. The field extension Q(ω) ⊂ Q(ω, ζ) is Galois with cyclic Galois group isomorphic to (Z/pZ)×. The unique field between Q(ω) and Q(ω, ζ) having degree m over Q(ω) takes the form Q(ω, τ) where τ is a Gauss sum, to be described below. Furthermore, under some conditions we can compute τ as an element α of Q(ω), thus expressing the degree-m intermediate field extension as a radical extension, Q(ω, τ) = Q ( ω, m √ α ) .