= Finite p-group embedding in the Nottingham group
Realize a <finite p-group> $P$ as the <Galois group> of a <totally ramified extension> $E/\mathbb F_p((t))$, using <Artin-Schreier embedding problems over a characteristic-p field>. A <uniformizer> $u$ of $E$ gives $E\cong\mathbb F_p((u))$: every element has a unique convergent Laurent expansion with coefficients in the <residue field>. An automorphism has $u\mapsto a_1u+a_2u^2+\cdots$, and $a_1$ defines a <homomorphism> $P\to\mathbb F_p^\times$. Its image is trivial because $|P|$ is a $p$-power and $p\nmid|\mathbb F_p^\times|$. Thus the faithful Galois action lies in the <Nottingham group>, embedding $P$.
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