 An element "g" of the Galois group "G" sends "ζ""n" to another primitive "p""n" root of unity

where "a""g","n" ∈ (Z/"p""n")×.For a given "g", as "n" varies, the "a""g","n" form a comptatible system in the sense that they give an element of the inverse limit of the (Z/"p""n")×, which is Zp×.Therefore, the "p"-adic cyclotomic character sends "g" to the system ("a""g","n")"n", thus encoding the action of "g" on all "p"-power roots of unity.