I am reading this lemma of Abstract Algebra by Dummit and Foote page 628. The only line that I couldn't understand this -5 line. Is $F'/F$ cyclic? As far as I understood, $F'$ is an extension over F, adjoining all needed roots of unit. But if $\boldsymbol{\zeta}_{n}$ denote a generator of multiplicative group of nth root of unity, which is cyclic, it is not always true that $Gal(F(\boldsymbol{\zeta}_{n})/F)$ cyclic, isn't it?
