= Artin-Schreier embedding problems over a characteristic-p field
{c}
For a <field> $F$ of characteristic $p$, the <exact sequence> $0\to\mathbb F_p\to F^{\mathrm{sep}}\xrightarrow{z\mapsto z^p-z}F^{\mathrm{sep}}\to0$, together with additive normal-basis acyclicity, gives $H^2(\operatorname{Gal}(F^{\mathrm{sep}}/F),\mathbb F_p)=0$. Consequently every central finite embedding problem with <group homomorphism kernel> $C_p$ has a continuous weak solution: its pulled-back <two-cocycle> is a <coboundary>, whose correcting one-cochain supplies the lift. Over $\mathbb F_p((t))$, independent Artin–Schreier classes $t^{-m}$, with $m>0$ prime to $p$, allow the lift to be twisted by characters so that inertia maps onto the <group homomorphism kernel> as well as onto the original quotient. Induction through central order-$p$ quotients therefore realizes every <finite p-group> as the <Galois group> of a <totally ramified extension> with <residue field> $\mathbb F_p$.
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