In number theory, the Teichmüller character
ω
, 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
-adic integers with the corresponding ones in the complex numbers,
can be considered as a usual Dirichlet character of conductor
More generally, given a complete discrete valuation ring
whose residue field
is perfect of characteristic
, there is a unique multiplicative section
of the natural surjection
The image of an element under this map is called its Teichmüller representative.
is called the Teichmüller character.
is the unique solution of
It can also be defined by The multiplicative group of
-adic units is a product of the finite group of roots of unity and a group isomorphic to the
The finite group is cyclic of order
is odd or even, respectively, and so it is isomorphic to
[citation needed] The Teichmüller character gives a canonical isomorphism between these two groups.
A detailed exposition of the construction of Teichmüller representatives for the
-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.