I was reading through Hensel's lemma, highlighted here on page $8$ and had a couple of queries:
In the proof when the author wants to show that $f(\alpha)=0$; it says something like "since $f(a_n)\equiv 0\bmod p^n$, we have that $\vert f(a_n)\vert_p\leq1/p^n$. Why is this the case? I'm not too sure as to where that's come from.
Also suppose we wish to find a $3$-adic expansion of $\sqrt{-1}$. Looking at $x^2+1\equiv 0\bmod 3$, we see that this has no roots. Does this mean that $\sqrt{-1}\in\Bbb Z_3$ because there are no roots?
In a similar token, suppose we wish to find a $5$-adic expansion of $\sqrt{-1}$. Then $x^2+1\equiv 0\bmod 5$ has roots and can be lifted as the derivative is non-zero at both roots. Then what does the $5$-adic expansion start with? In other words, in the power series representation, what would the "$5^0$ term" be?
Finally, is my notation right? Like does Hensel's lemma look at $\Bbb Z_p$ or $\Bbb Q_p$?
Thanks in advance!