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Teichmüller character

Special character in number theory

Summary

Special character in number theory

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^\times is called the Teichmüller character.

Definition

If x is an integer mod p, 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. 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 | author-link=Henri Cohen (number theorist)

Wikipedia Source

This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.

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