Editing Kummer theory (section)

== Generalizations ==

Suppose that ''G'' is a [[profinite group]] acting on a module ''A'' with a surjective homomorphism π from the ''G''-module ''A'' to itself. Suppose also that ''G'' acts trivially on the kernel ''C'' of π and that the first cohomology group H<sup>1</sup>(''G'',''A'') is trivial. Then the exact sequence of group cohomology shows that there is an isomorphism between ''A''<sup>''G''</sup>/π(''A''<sup>''G''</sup>) and Hom(''G'',''C'').

Kummer theory is the special case of this when ''A'' is the multiplicative group of the separable closure of a field ''k'', ''G'' is the Galois group, π is the ''n''th power map, and ''C'' the group of ''n''th roots of unity. [[Artin–Schreier theory]] is the special case when ''A'' is the additive group of the separable closure of a field ''k'' of positive characteristic ''p'', ''G'' is the Galois group, π is the [[Frobenius map]] minus the identity, and ''C'' the finite field of order ''p''. Taking ''A'' to be a ring of truncated Witt vectors gives Witt's generalization of Artin–Schreier theory to extensions of exponent dividing ''p<sup>n</sup>''.