Existence/uniqueness of the Teichmuller map for p-adic integers

Question

Let $Z_p$ denote the p-adic integers. Let $π:Z_p \to \mathbb{Z}/p\mathbb{Z}$ by $π(a_0+a_1p+a_2p^2+...)=a_0$ where, of course, each $a_i \in \{0,1,...,p-1\}$

According to my professor, there exists some unique function $T:\mathbb{Z}/p\mathbb{Z}\to Z_p$ satisfying the following two properties:

1. $\forall x \in \mathbb{Z}/p\mathbb{Z}[π(T(x))=x]$

2. $\forall x,y \in \mathbb{Z}/p\mathbb{Z}[T(x)^p=T(x)]$

My question is, how can we show that $T$ really exists and is really unique? Also, would it be a homomorphism or not?

Answer 1

Hint: $Z_p$ is an integral domain so that $T(x)^p = T(x)$ for all $x$ implies that either $T(x) = 0$ or $T(x)^{p-1} = 1$ for any $x \in \mathbb Z/p\mathbb Z$. So this is a really strong condition on the images of $T(x)$. Using this and the first condition will give you the construction, and it's uniqueness with some work.

It will not be an additive or ring homomorphism, as $\mathbf Z/p\mathbf Z$ is $p$-torsion additively and $Z_p$ has no torsion. It will however be multiplicative, in the sense that $T(xy) = T(x)T(y)$.

Answer 2

For existence, I like the trick of forming, for any integer $m$, the sequence $z_0=m$, $z_{n+1}=z_n^p$. You need a little lemma to show that the sequence is convergent, but clearly the limit $z_\infty$ satisfies $z_\infty^p=z_\infty$. Start with $m$ prime to $p$, and get a root of unity.

For uniqueness, I think you may be able to analyse the above construction to get that.