Teichmüller character

In number theory, the Teichmüller character ω (at a prime p) is a character of (Z/qZ)^×, 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 p-adic integers with the corresponding ones in the complex numbers, ω can be considered as a usual Dirichlet character of conductor q. More generally, given a complete discrete valuation ring O whose residue field k is perfect of characteristic p, there is a unique multiplicative section ω : k → O of the natural surjection O → k. The image of an element under this map is called its Teichmüller representative. The restriction of ω to k^× is called the Teichmüller character.