Artin–Schreier theory

《Local Fields》 · 第 154–155 页 · Chapter X，Section 3(a)「Artin-Schreier Theory」

(a) Artin-Schreier Theory. Let $k$ be a field of characteristic $p\ne0$, let $K$ be its separable algebraic closure, and let $G=\operatorname{Gal}(K/k)$. The map $\wp:K\to K$, $\wp(x)=x^p-x$, is surjective and has kernel $\mathbb Z/p\mathbb Z$, hence gives an exact sequence $0\to\mathbb Z/p\mathbb Z\to K\xrightarrow{\wp}K\to0$ and an isomorphism $H^1(G,\mathbb Z/p\mathbb Z)\simeq k/\wp(k)$.