Teichmüller character

In number theory, the Teichmüller character ω (at a prime p) is a character of (Z/qZ)×, where $$q = p$$ if $$p$$ is odd and $$q = 4$$ if $$p = 2$$, 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, &omega; 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.

Definition
If x is a p-adic integer, then $$\omega(x)$$ is the unique solution of $$\omega(x)^p = \omega(x)$$ that is congruent to x mod p. It can also be defined by


 * $$\omega(x)=\lim_{n\rightarrow\infty} x^{p^n}$$

The multiplicative group of p-adic units is a product of the finite group of roots of unity and a group isomorphic to the p-adic integers. The finite group is cyclic of order p – 1 or 2, as p is odd or even, respectively, and so it is isomorphic to (Z/qZ)×. The Teichmüller character gives a canonical isomorphism between these two groups.

A detailed exposition of the construction of Teichmüller representatives for the p-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.