p-adic numbers and $\mathbb{F}_p$

Question

As you probably know, there is a morphism of rings : $\mathbb{Z}_p\longrightarrow \mathbb{Z}/p\mathbb{Z}$ which sends a formal sum $\sum_{i\geq 0}a_ip^i$ to $a_0$ (here $\mathbb{Z}_p$ is the ring of $p$-adic numbers). %y question is : is there a morphism of rings $\mathbb{Z}/p\mathbb{Z}\longrightarrow \mathbb{Z}_p$ ?

Answer 1

There is not. In $\mathbb{Z} / p \mathbb{Z}$ we have the identity $p=0$. However, in $\mathbb{Z}_p$, $p \neq 0$. However, any homomorphism of rings has to send $0$ to $0$ and $p$ to $p$.

Answer 2

Exercise: Show that if $R\to S$ is a ring homomorphism (and $S$ is an integral domain), then $({\rm char}~S)\mid({\rm char}~R)$. Conclude there is no homomorphism ${\bf F}_p\to{\bf Z}_p$. (This is Hurkyl's answer.)

However. Every element of ${\bf F}_p^\times$ is a $(p-1)$th root of unity, and there is a subgroup $\mu_{p-1}\le {\bf Z}_p^\times$ consisting of these roots of unity in the $p$-adic integers (one may establish their existence using Hensel's lemma). Thus there is a multiplicative group homomorphism $\tau:{\bf F}_p^\times\to{\bf Z}_p^\times$. If we add the stipulation that $\tau(0)=0$, its image consists of a set of representatives for ${\bf F}_p\cong {\bf Z}_p/p{\bf Z}_p$.

When we try to verify the additive part of ring homomorphisms on our map $\tau$, we find that $\tau(x+y)\ne\tau(x)+\tau(y)$; there is a discrepancy between addition before and after. We will see it is possible to patch up this gap so that we do obtain a ring homomorphism (isomorphism, in fact), but we must expand the codomain with extra packets of information, wherein each copy of ${\bf F}_p$ we tack on will patch up part of the gap left between the copies we had before and the full ring ${\bf Z}_p$.

Exercise: Let $X$ be a set of coset representatives of ${\bf Z}_p/p{\bf Z}_p$ within ${\bf Z}_p$. Then every element of ${\bf Z}_p$ may be written uniquely as $p$-adic expansion with 'digits' taken from $X$. (Hint: construct the sequence of digits iteratively, so that the partial sums converge to a given element in the limit.)

Thus we may identify the underlying sets of ${\bf Z}_p$ and ${\bf F}_p^{\bf N}$; if we could encode the addition and multipliction operations in ${\bf Z}_p$ by how the appear when transported onto the space ${\bf F}_p^{\bf N}$, we will have achieved a task spiritually overarching our original want for a homomorphism ${\bf F}_p\to {\bf Z}_p$. Our map $\tau$ may be thought of as an "almost-homomorphism," and the successive approximations may be thought of as the inverse map of truncating expansions ${\bf Z}_p\to {\bf F}_p^n$ as $n\to\infty$.

This is where the theory of Teichmüller characters / representatives (i.e. our $\tau$) and of Witt vectors comes into play, allowing us to construct rings from residue fields. An elevated view of the territory:

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Taken from notes The Theory of Witt Vectors by Rabinoff. (Online pdf.)