Teichmüller character
Corollary.
Let p be a prime number. The ring of p-adic integers (http://planetmath.org/PAdicIntegers) Zp contains exactly p-1 distinct (p-1)th roots of unity
. Furthermore, every (p-1)th root of unity is distinct modulo p.
Proof.
Notice that ℚp, the p-adic rationals, is a field. Therefore f(x)=xp-1-1 has at most p-1 roots in ℚp (see this entry (http://planetmath.org/APolynomialOfDegreeNOverAFieldHasAtMostNRoots)). Moreover, if we let a∈ℤ with 1≤a≤p-1 then f(a)=ap-1-1≡0mod by Fermat’s little theorem. Since is non-zero modulo , the trivial case of Hensel’s lemma implies that there exist a root of in which is congruent to modulo . Hence, there are at least roots in , and we can conclude that there are exactly roots.
∎
Definition.
The Teichmüller character is a homomorphism of multiplicative groups
:
such that is the unique th root of unity in which is congruent to modulo (which exists by the corollary above). The map is sometimes called the Teichmüller lift of to ( would lift to ).
Remark.
Some authors define the Teichmüller character to be the homomorphism:
defined by
Notice that for any , is a th root of unity:
Thus, the value is the same than .
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 |