Teichmüller character

In number theory, the Teichmüller character $\omega$ (at a prime $p$) is a character of $(\mathbb {Z} /q\mathbb {Z} )^{\times }$, 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 $(\mathbb {Z} /q\mathbb {Z} )^{\times }$. 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 Cohen, Henri (2007), Number theory, Volume I: Tools and Diophantine equations, Graduate Texts in Mathematics, vol. 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