Teichmüller character

In number theory, the Teichmüller character ${\displaystyle \omega }$ (at a prime ${\displaystyle p}$) is a character of ${\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{\times }}$, where ${\displaystyle q=p}$ if ${\displaystyle p}$ is odd and ${\displaystyle q=4}$ if ${\displaystyle p=2}$, taking values in the roots of unity of the p-adic integers. It was introduced by Oswald Teichmüller. Identifying the roots of unity in the ${\displaystyle p}$-adic integers with the corresponding ones in the complex numbers, ${\displaystyle \omega }$ can be considered as a usual Dirichlet character of conductor ${\displaystyle q}$. More generally, given a complete discrete valuation ring ${\displaystyle O}$ whose residue field ${\displaystyle k}$ is perfect of characteristic ${\displaystyle p}$, there is a unique multiplicative section ${\displaystyle \omega :k\to O}$ of the natural surjection ${\displaystyle O\to k}$. The image of an element under this map is called its Teichmüller representative. The restriction of ${\displaystyle \omega }$ to ${\displaystyle k^{x}}$ is called the Teichmüller character.