Kummer extension

Suppose $n$ is prime to the characteristic of a field $K$ and $K$ contains all $n$th roots of unity. For $a\in K^{\times }$, adjoining a root of $X^n-a$ gives a Kummer extension. Every conjugate of a chosen root $\alpha$ is $\zeta \alpha$ with $\zeta \in {\mu }_n\subseteq K$, so the extension is a cyclic Galois extension of degree dividing $n$.

Adjoining several such radicals gives the finite abelian extensions of exponent dividing $n$ described by Kummer theory. The characteristic restriction is essential: in characteristic $p$, adjoining a $p$th root of an element that is not a $p$th power produces a purely inseparable extension, with no nonidentity automorphisms. Cyclic extensions of degree $p$ in that characteristic are described instead by Artin–Schreier theory.