In number theory, the Teichmüller character ω {\displaystyle \omega } (at a prime p {\displaystyle p} ) is a character of ( Z / q Z ) × {\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{\times }} , where q = p {\displaystyle q=p} if p {\displaystyle p} is odd and q = 4 {\displaystyle q=4} if p = 2 {\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 p {\displaystyle p} -adic integers with the corresponding ones in the complex numbers, ω {\displaystyle \omega } can be considered as a usual Dirichlet character of conductor q {\displaystyle q} . More generally, given a complete discrete valuation ring O {\displaystyle O} whose residue field k {\displaystyle k} is perfect of characteristic p {\displaystyle p} , there is a unique multiplicative section ω : k → O {\displaystyle \omega :k\to O} of the natural surjection O → k {\displaystyle O\to k} . The image of an element under this map is called its Teichmüller representative. The restriction of ω {\displaystyle \omega } to k × {\displaystyle k^{\times }} is called the Teichmüller character.
Definition If x {\displaystyle x} is an integer mod p {\displaystyle p} , then ω ( x ) {\displaystyle \omega (x)} is the unique solution of ω ( x ) p = ω ( x ) {\displaystyle \omega (x)^{p}=\omega (x)} that is congruent to x {\displaystyle x} mod p {\displaystyle p} . It can also be defined by
ω ( x ) = lim n → ∞ x p n {\displaystyle \omega (x)=\lim _{n\rightarrow \infty }x^{p^{n}}}
The multiplicative group of p {\displaystyle p} -adic units is a product of the finite group of roots of unity and a group isomorphic to the p {\displaystyle p} -adic integers. The finite group is cyclic of order p − 1 {\displaystyle p-1} or 2 {\displaystyle 2} , as p {\displaystyle p} is odd or even, respectively, and so it is isomorphic to ( Z / q Z ) × {\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 p {\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
