Kummer–Artin–Schreier–Witt theory

Summary

Kummer–Artin–Schreier–Witt theory describes the abelian extensions of bounded exponent of a field by explicit equations. For an exponent $m$ prime to the characteristic, this is Kummer theory. It uses $m$-th roots and needs the $m$-th roots of unity in the base field. For an exponent $p^n$ in characteristic $p > 0$, this is Artin–Schreier–Witt theory. It uses equations $F(x) - x = a$ in the ring of Witt vectors of length $n$. For $n = 1$ it reduces to Artin–Schreier theory. Together the two parts describe all finite cyclic extensions of a field that contains enough roots of unity.