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In number theory, the Teichmüller character ${\displaystyle \omega }$ (at a prime ${\displaystyle p}$) is a character of ${\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{\times }}$, where ${\displaystyle q=p}$ if ${\displaystyle p}$ is odd and ${\displaystyle q=4}$ if ${\displaystyle p=2}$, 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 ${\displaystyle p}$-adic integers with the corresponding ones in the complex numbers, ${\displaystyle \omega }$ can be considered as a usual Dirichlet character of conductor ${\displaystyle q}$. More generally, given a complete discrete valuation ring ${\displaystyle O}$ whose residue field ${\displaystyle k}$ is perfect of characteristic ${\displaystyle p}$, there is a unique multiplicative section ${\displaystyle \omega :k\to O}$ of the natural surjection ${\displaystyle O\to k}$. The image of an element under this map is called its Teichmüller representative. The restriction of ${\displaystyle \omega }$ to ${\displaystyle k^{x}}$ is called the Teichmüller character.

Definition

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

${\displaystyle \omega (x)=\lim _{n\rightarrow \infty }x^{p^{n}}}$

The multiplicative group of ${\displaystyle p}$-adic units is a product of the finite group of roots of unity and a group isomorphic to the ${\displaystyle p}$-adic integers. The finite group is cyclic of order ${\displaystyle p-1}$ or ${\displaystyle 2}$, as ${\displaystyle p}$ is odd or even, respectively, and so it is isomorphic to ${\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{\times }}$. The Teichmüller character gives a canonical isomorphism between these two groups.

A detailed exposition of the construction of Teichmüller representatives for the ${\displaystyle p}$-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

• Class field theory