Kummer Theory

Kummer theory provides converse statements. When K contains n distinct nth roots of unity, it states that any cyclic extension of K of degree n is formed by extraction of an nth root. Further, if K× denotes the multiplicative group of non-zero elements of K, cyclic extensions of K of degree n correspond bijectively with cyclic subgroups of

that is, elements of K× modulo nth powers. The correspondence can be described explicitly as follows. Given a cyclic subgroup

the corresponding extension is given by

that is, by adjoining nth roots of elements of Δ to K. Conversely, if L is a Kummer extension of K, then Δ is recovered by the rule

In this case there is an isomorphism

given by

where α is any nth root of a in L.