Teichmüller character

Before we define the Teichmüller characterPlanetmathPlanetmath, we begin with a corollary of Hensel’s lemma.


Let p be a prime numberMathworldPlanetmath. The ring of p-adic integers (http://planetmath.org/PAdicIntegers) Zp contains exactly p-1 distinct (p-1)th roots of unityMathworldPlanetmath. Furthermore, every (p-1)th root of unity is distinct modulo p.


Notice that p, the p-adic rationals, is a field. Therefore f(x)=xp-1-1 has at most p-1 roots in p (see this entry (http://planetmath.org/APolynomialOfDegreeNOverAFieldHasAtMostNRoots)). Moreover, if we let a with 1ap-1 then f(a)=ap-1-10modp by Fermat’s little theorem. Since f(a)=(p-1)ap-2 is non-zero modulo p, the trivial case of Hensel’s lemma implies that there exist a root of xp-1-1 in p which is congruentMathworldPlanetmath to a modulo p. Hence, there are at least p-1 roots in p, and we can conclude that there are exactly p-1 roots. ∎


The Teichmüller character is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of multiplicative groupsMathworldPlanetmath:


such that ω(a) is the unique (p-1)th root of unity in Zp which is congruent to a modulo p (which exists by the corollary above). The map ω is sometimes called the Teichmüller lift of Fp to Zp (0modp would lift to 0Zp).


Some authors define the Teichmüller character to be the homomorphism:


defined by


Notice that for any zp×, ω^(z) is a (p-1)th root of unity:


Thus, the value ω^(z) is the same than ω(zmodp).

Title Teichmüller character
Canonical name TeichmullerCharacter
Date of creation 2013-03-22 15:09:04
Last modified on 2013-03-22 15:09:04
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 7
Author alozano (2414)
Entry type Definition
Classification msc 13H99
Classification msc 11S99
Classification msc 12J99
Synonym Teichmuler character
Synonym Teichmuller lift
Synonym Teichmüller lift
Related topic PAdicIntegers