# Teichmüller character

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

###### Corollary.

Let $p$ be a prime number. The ring of $p$-adic integers (http://planetmath.org/PAdicIntegers) $\mathbb{Z}_{p}$ 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 $\mathbb{Q}_{p}$, the $p$-adic rationals, is a field. Therefore $f(x)=x^{p-1}-1$ has at most $p-1$ roots in $\mathbb{Q}_{p}$ (see this entry (http://planetmath.org/APolynomialOfDegreeNOverAFieldHasAtMostNRoots)). Moreover, if we let $a\in\mathbb{Z}$ with $1\leq a\leq p-1$ then $f(a)=a^{p-1}-1\equiv 0\mod p$ by Fermat’s little theorem. Since $f^{\prime}(a)=(p-1)\cdot a^{p-2}$ is non-zero modulo $p$, the trivial case of Hensel’s lemma implies that there exist a root of $x^{p-1}-1$ in $\mathbb{Z}_{p}$ which is congruent to $a$ modulo $p$. Hence, there are at least $p-1$ roots in $\mathbb{Z}_{p}$, and we can conclude that there are exactly $p-1$ roots. ∎

###### Definition.

The Teichmüller character is a homomorphism of multiplicative groups:

 $\omega\colon\mathbb{F}_{p}^{\times}\to\mathbb{Z}_{p}^{\times}$

such that $\omega(a)$ is the unique $(p-1)$th root of unity in $\mathbb{Z}_{p}$ which is congruent to $a$ modulo $p$ (which exists by the corollary above). The map $\omega$ is sometimes called the Teichmüller lift of $\mathbb{F}_{p}$ to $\mathbb{Z}_{p}$ ($0\mod p$ would lift to $0\in\mathbb{Z}_{p}$).

###### Remark.

Some authors define the Teichmüller character to be the homomorphism:

 $\hat{\omega}\colon\mathbb{Z}_{p}^{\times}\to\mathbb{Z}_{p}^{\times}$

defined by

 $\hat{\omega}(z)=\lim_{n\to\infty}z^{p^{n}}.$

Notice that for any $z\in\mathbb{Z}_{p}^{\times}$, $\hat{\omega}(z)$ is a $(p-1)$th root of unity:

 $(\hat{\omega}(z))^{p}=\left(\lim_{n\to\infty}z^{p^{n}}\right)^{p}=\lim_{n\to% \infty}z^{p^{n+1}}=\hat{\omega}(z).$

Thus, the value $\hat{\omega}(z)$ is the same than $\omega(z\mod p)$.

 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