Teichmüller character

In number theory, the Teichmüller character $\omega$ (at a prime $p$ ) is a character of $(\Z/q\Z)^\times$ , 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 $\omega:k\to O$ of the natural surjection $O\to k$ . The image of an element under this map is called its Teichmüller representative. The restriction of $\omega$ to $k^x$ 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 $(\Z/q\Z)^\times$ .{{fact|reason=And this is violated when p is 2 and q is 4?|date=May 2014}} 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.

References

Section 4.3 of {{Citation

| last=Cohen

| first=Henri

| author-link=Henri Cohen (number theorist)

| title=Number theory, Volume I: Tools and Diophantine equations

| publisher=Springer

| location=New York

| series=Graduate Texts in Mathematics

| volume=239

| year=2007

| isbn=978-0-387-49922-2

| mr=2312337

| doi=10.1007/978-0-387-49923-9

}}

{{Citation | last1=Koblitz | first1=Neal | author1-link=Neal Koblitz | title=p-adic Numbers, p-adic Analysis, and Zeta-Functions | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics, vol. 58 | isbn=978-0-387-96017-3 | mr=754003 | year=1984}}

Category:Class field theory

Teichmüller character

Definition

See also

References