# Teichmüller character

In number theory, the **Teichmüller character** ω (at a prime *p*) is a character of (**Z**/*q***Z**)^{×}, 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 *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**.

## Definition

If *x* is a *p*-adic integer, then is the unique solution of that is congruent to *x* mod *p*. It can also be defined by

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**/*q***Z**)^{×}. 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

## References

- Section 4.3 of Cohen, Henri (2007),
*Number theory, Volume I: Tools and Diophantine equations*, Graduate Texts in Mathematics,**239**, New York: Springer, doi:10.1007/978-0-387-49923-9, ISBN 978-0-387-49922-2, MR 2312337

- Koblitz, Neal (1984),
*p-adic Numbers, p-adic Analysis, and Zeta-Functions*, Graduate Texts in Mathematics, vol. 58, Berlin, New York: Springer-Verlag, ISBN 978-0-387-96017-3, MR 0754003