Teichmüller character explained

In number theory, the Teichmüller character

\omega (at a prime p ) is a character of (\Z/q\Z) ^{x} , 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 ^{x} 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 → 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) ^{x} . 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.

See also

• Witt vector

References

• Section 4.3 of