Cyclic Stickelberger cohomology and descent of Kummer extensions

Abstract

Let R R be a field, S = R [ ζ ] S = R[{\rm {\zeta }}] , ζ {\rm {\zeta }} an n n th root of unit, Δ = G a l ( S / R ) \Delta = {\rm {Gal(}}S/R) . The group of cyclic Kummer extensions of S S on which Δ \Delta acts, modulo those which descend to R R , is isomorphic to a group of roots of unity and to a second group cohomology group of Δ \Delta whose definition involves a "Stickelberger element".