{{Short description|Special character in number theory}} In number theory, the '''Teichmüller character''' <math>\omega</math> (at a prime <math>p</math>) is a character of <math>(\Z/q\Z)^\times</math>, where <math>q = p</math> if <math>p</math> is odd and <math>q = 4</math> if <math>p = 2</math>, 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 <math>p</math>-adic integers with the corresponding ones in the complex numbers, <math>\omega</math> can be considered as a usual Dirichlet character of conductor <math>q</math>. More generally, given a complete discrete valuation ring <math>O</math> whose residue field <math>k</math> is perfect of characteristic <math>p</math>, there is a unique multiplicative section <math>\omega:k\to O</math> of the natural surjection <math>O\to k</math>. The image of an element under this map is called its '''Teichmüller representative'''. The restriction of <math>\omega</math> to <math>k^\times</math> is called the '''Teichmüller character'''.
==Definition==
If <math>x</math> is an integer mod <math>p</math>, then <math>\omega(x)</math> is the unique solution of <math>\omega(x)^p = \omega(x)</math> that is congruent to <math>x</math> mod <math>p</math>. It can also be defined by
:<math>\omega(x)=\lim_{n\rightarrow\infty} x^{p^n}</math>
The multiplicative group of <math>p</math>-adic units is a product of the finite group of roots of unity and a group isomorphic to the <math>p</math>-adic integers. The finite group is cyclic of order <math>p-1</math> or <math>2</math>, as <math>p</math> is odd or even, respectively, and so it is isomorphic to <math>(\Z/q\Z)^\times</math>.{{fact|reason=And this is violated when p is 2 and q is 4?|date=May 2014}} The Teichmüller character gives a canonical isomorphism between these two groups.
A detailed exposition of the construction of Teichmüller representatives for the <math>p</math>-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 {{Citation | last=Cohen | first=Henri | author-link=Henri Cohen (number theorist) | title=Number theory, Volume I: Tools and Diophantine equations | publisher=Springer | location=New York | series=Graduate Texts in Mathematics | volume=239 | year=2007 | isbn=978-0-387-49922-2 | mr=2312337 | doi=10.1007/978-0-387-49923-9 }}
*{{Citation | last1=Koblitz | first1=Neal | author1-link=Neal Koblitz | title=p-adic Numbers, p-adic Analysis, and Zeta-Functions | publisher=Springer-Verlag | location=Berlin, New York | series=Graduate Texts in Mathematics, vol. 58 | isbn=978-0-387-96017-3 | mr=754003 | year=1984}}
{{DEFAULTSORT:Teichmuller character}} Category:Class field theory