What is a Kummer extension?
Solution
Let \(K\) be a field, let \(n\ge2\) be an integer with \[ \operatorname{char}K\nmid n, \] and suppose \(K\) contains the group \(\mu_n\) of \(n\)th roots of unity.
A Kummer extension of exponent dividing \(n\) is an extension of the form \[ L=K\bigl(\sqrt[n]{a_1},\ldots,\sqrt[n]{a_r}\bigr) \] for elements \(a_i\in K^\times\), with \(L/K\) finite.
Because \(K\) contains \(\mu_n\), each polynomial \[ x^n-a_i \] splits over \(L\) once one root is present. Hence \(L/K\) is normal. Since \(\operatorname{char}K\nmid n\), these polynomials are separable, so \(L/K\) is Galois.
Every automorphism sends \[ \sqrt[n]{a_i}\longmapsto \zeta_i\sqrt[n]{a_i} \qquad(\zeta_i\in\mu_n), \] so the Galois group embeds into a finite product of copies of \(\mu_n\). Therefore it is abelian and has exponent dividing \(n\).
Conversely, Kummer theory says that every finite abelian Galois extension \(L/K\) whose Galois group has exponent dividing \(n\) is obtained in this way, under the same hypotheses on \(K\).