Teichmüller character explained

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, 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 of the natural surjection . 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 → 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.

See also

• Witt vector

References

• Section 4.3 of