Teichmüller character

In number theory, the Teichmüller character is a character of, where if is odd and if, 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 -adic integers with the corresponding ones in the complex numbers, can be considered as a usual Dirichlet character of conductor. More generally, given a complete discrete valuation ring whose residue field is perfect of characteristic, there is a unique multiplicative section of the natural surjection. The image of an element under this map is called its Teichmüller representative. The restriction of to is called the Teichmüller character.

Definition

If is an integer mod, then is the unique solution of that is congruent to mod. It can also be defined by
The multiplicative group of -adic units is a product of the finite group of roots of unity and a group isomorphic to the -adic integers. The finite group is cyclic of order or, as is odd or even, respectively, and so it is isomorphic to. The Teichmüller character gives a canonical isomorphism between these two groups.
A detailed exposition of the construction of Teichmüller representatives for the -adic integers, by means of Hensel lifting, is given in the article on Witt vectors, where they provide an important role in providing a ring structure.