Kummer extension

A Kummer extension is what you build by taking n-th roots, in the lucky situation where your base field already contains all the n-th roots of unity. Adjoining the n-th root of an element then turns out to be remarkably well-behaved: the symmetry group is abelian, and you can read it off directly from which elements you took roots of.

Precisely, suppose K is a field containing a primitive n-th root of unity (so n is invertible in K). A Kummer extension of exponent n is a field L = K(a_1^{1/n}, ..., a_r^{1/n}) obtained by adjoining n-th roots of finitely many elements a_i in K. Such an L is a finite abelian extension of K whose Galois group has exponent dividing n. Kummer theory makes this an exact correspondence: abelian extensions of exponent n of K are in bijection with subgroups of K^* / (K^*)^n that contain (K^*)^n, the subgroup being generated by the a_i.

The hypothesis that the roots of unity are already present is essential and is the whole point. Without it — for example adjoining the real cube root of 2 to Q, which lacks the primitive cube roots of unity — the extension is not even Galois, and the clean abelian description collapses. Kummer theory is the converse companion to the structure theorem: every cyclic extension of degree n of a field with enough roots of unity arises by adjoining a single n-th root.