Cyclic extensions of prime order $p$

Let \(K/F\) be cyclic of prime degree \(p\).

If \(\operatorname{char}F\ne p\) and \(F\) contains a primitive \(p\)th root of unity \(\zeta_p\), Kummer theory gives \[ K=F(\alpha),\qquad \alpha^p=a\in F^\times, \] with \(a\notin F^{\times p}\). A generator of the Galois group acts by \[ \sigma(\alpha)=\zeta_p\alpha. \] The extension depends only on the one-dimensional \(\mathbb F_p\)-subspace generated by the class of \(a\) in \[ F^\times/F^{\times p}. \] If \(\mu_p\not\subset F\), the corresponding classification is obtained after adjoining \(\zeta_p\) together with the usual descent condition; there is no equally simple description by adjoining a \(p\)th root of an element of \(F\) itself.

If \(\operatorname{char}F=p\), Artin–Schreier theory gives \[ K=F(\alpha),\qquad \alpha^p-\alpha=a\in F, \] where \(a\notin\wp(F)\) and \[ \wp(x)=x^p-x. \] A generator acts by \[ \sigma(\alpha)=\alpha+1. \] The extension depends on the one-dimensional \(\mathbb F_p\)-subspace generated by the class of \(a\) in \[ F/\wp(F). \] These are the Kummer and Artin–Schreier classifications of cyclic extensions of prime degree.