Teichmüller character
Encyclopedia
In number theory
, the Teichmüller character ω (at a prime p) is a character of (Z/qZ)×, where q = p or 4 if p odd or even, 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
of the natural surjection . The image of an element under this map is called its Teichmüller representative. The restriction of ω to k× is called the Teichmüller character.
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.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, the Teichmüller character ω (at a prime p) is a character of (Z/qZ)×, where q = p or 4 if p odd or even, taking values in the roots of unity of the p-adic integers. It was introduced by Oswald Teichmüller
Oswald Teichmüller
Oswald Teichmüller was a German mathematician who introduced quasiconformal mappings and differential geometric methods into complex analysis.-Life:...
. 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
Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z...
of conductor q. More generally, given a complete discrete valuation ring
Discrete valuation ring
In abstract algebra, a discrete valuation ring is a principal ideal domain with exactly one non-zero maximal ideal.This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:...
O whose residue field k is perfect
Perfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:* Every irreducible polynomial over k has distinct roots.* Every polynomial over k is separable.* Every finite extension of k is separable...
of characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
p, there is a unique multiplicative section
Section (category theory)
In category theory, a branch of mathematics, a section is a right inverse of a morphism. Dually, a retraction is a left inverse...
of the natural surjection . 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 ω(x) is the unique solution of ω(x)p = ω(x) that is congruent to x mod p. It can also be defined byThe 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.