Generalizations
There exists a slight generalization of Kummer theory which deals with abelian extensions with Galois group of exponent n, and an analogous statement is true in this context. Namely, one can prove that such extensions are in one-to-one correspondence with subgroups of
If the ground field K does not contain the nth roots of unity, one sometimes still uses the phrase Kummer theory to refer to the isomorphism
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