= Finite coefficient field in positive-characteristic local fields
{title2=$\mathbb F_q\hookrightarrow\mathcal O_K,\qquad a^q=a$}
For a characteristic-p <local field> with <residue field> $\mathbb F_q$, the simple roots of $X^q-X$ lift uniquely by <Hensel lemma>. Their lifts form a copy of $\mathbb F_q$, since qth powers preserve addition and multiplication. Choose a <uniformizer> $\pi$. Repeatedly subtracting the lifted residue and dividing by $\pi$ produces a unique convergent expansion $\sum_{i\ge0}a_i\pi^i$ for each integral element. Allowing finitely many negative powers gives a valued-field <isomorphism> $K\cong\mathbb F_q((T))$, $T\mapsto\pi$.
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