= Kummer extension
{c}
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>.
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