Teichmüller character

The Teichmuller character sends a nonzero residue class to its unique root-of-unity lift in the valuation ring.

The Teichmuller representative of a nonzero residue class $\bar{a}\in {\mathbb{F}}_q^{\ast }$ is the unique lift $\omega$ satisfying ${\omega }^{q-1}=1$ and $\omega \equiv a$ modulo the maximal ideal.

For a prime $p$ and $n\ge 1$, the unique residue $y$ modulo $p^n$ satisfying $y\equiv x(\mathrm{mod}p)$ and $y^p\equiv y(\mathrm{mod}{p}^n)$ is the displayed residue. Existence follows from Fermat's little theorem and prime-power contraction under pth powers. Uniqueness follows by raising any other such residue repeatedly to the $p$th power and applying the same contraction to its congruence with $x$. This is the finite-modulus version of a Teichmuller representative, including the zero class.

For $K/{\mathbb{Q}}_p$ finite, let $x_n^{p^n}=x$ in its finite residue field, and choose arbitrary lifts $y_n\in {\mathcal{O}}_K$. With normalized discrete valuation,Applying this binomial estimate repeatedly to $y_n$ and $[x_n]$ proves the displayed bound, and hence convergence to $[x]$.

For residue field ${\mathbb{F}}_q$, these two properties characterize the Teichmuller lift on every residue class, including zero. The Hensel lemma applies to $T^q-T$, whose derivative reduces to $-1$. Uniqueness implies $[ab]=[a][b]$, $[0]=0$, $[1]=1$, and $[a^p]=[a{]}^p$. In mixed characteristic this multiplicative section need not be additive.

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