Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $\mathbb{Z}/p\mathbb{Z}\longrightarrow \mathbb{Z}_p$ ?

share|improve this question

2 Answers 2

up vote 12 down vote accepted

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 $p$.

share|improve this answer

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:

$\quad$ witt vectors

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

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.