Teichmüller character

In number theory, the Teichmüller character ω (at a prime p) is a character of (Z/qZ)^×, 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, 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 $\omega (x)$ 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/qZ)^×. 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.

gollark: It *is* pretty javan.

gollark: Like I said, you should make a compiler thing to "warp" stuff for you!

gollark: Interesting.

gollark: You said you could probably do infinite storage, so I assume it's doable maybe ish.

gollark: You could show that it *was* TC if you implement brain[REDACTED] or another Turing complete language in it.

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

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

Teichmüller character

Definition

See also

References