I am reading Serre's A Course in Arithmetic and am having trouble understanding what he means by a polynomial deduced from a polynomial over $\mathbb{Z}_p$.
Specifically Serre writes,
Notation.-- If $f\in\mathbb{Z}_p[X_1,\dots,X_m]$ is a polynomial with coefficients in $\mathbb{Z}_p$, and if $n$ is an integer $\geq 1$, we denote by $f_n$ the polynomial with coefficients in $A_n$ deduced from $f$ by reduction $(\bmod p^n)$.
Now, given Serre's definition of an element of $\mathbb{Z}_p$ as a sequence of elements of successive $A_n=\mathbb{Z}/p^n\mathbb{Z}$, $n\geq 1$, I find imagining $f$ somewhat tricky in its own right. However, I am not sure what is meant by $f\bmod p^n$; is this $f$ where each coefficient is considered modulo $p^n$, in which case what does it mean to consider a sequence $(\dots, x_k, x_{k-1},\dots, x_1)$ modulo an integer?
From the definition of the $p$-adic numbers, you get a natural map $\mathbb Z_p \to \mathbb Z/p^n\mathbb Z$ for all $n$. We can call this map modulo $p^n$.
For example, if you have $a=(1,1,10,64,...)$ in $\mathbb Z_3$, then $a \mod 3=1$, $a \mod 9=1$, $a \mod 27=10$, $a \mod 81=64$.