Teichmuller expansion

If $K$ is a complete discretely valued field whose residue field $k$ is a perfect field, and $\pi$ is a uniformizer, the multiplicative Teichmuller lift $[\cdot ]:k\to {\mathcal{O}}_K$ gives each $x\in {\mathcal{O}}_K$ a unique expansion

If a complete discretely valued field $K$ has positive characteristic and perfect residue field $k$, its Teichmuller lift is the unique ring section $k\to {\mathcal{O}}_K$. A choice of uniformizer $t$ then gives

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 ${\sum }_{i\ge 0}{a}_i{\pi }^i$ for each integral element. Allowing finitely many negative powers gives a valued-field isomorphism $K\cong {\mathbb{F}}_q((T))$, $T\mapsto \pi$.