The problem is: find how many roots in $\mathbb{Q}_{p}$ does $x^{3}+25x^{2}+x-9$ have for $p=2,3,5,7$.
I found for $\mathbb{Z}_{p}$. Is there a way to extend from here?
Also, can you suggest me a book or link on this topic (local fields, roots of polynomials etc);I don't find my current book helpful.
On
To expand on both @RagibZaman’s comment and @DanielMiller’s answer, the result is true for any Unique Factorization Domain; and the proof is very elementary. Suppose $R$ is a UFD, and $F(X)\in R[X]$ with highest coefficient being $1$, so $F=X^m+c_{m-1}X^{m-1}+\cdots+c_1X+c_0$. Suppose $\alpha=r/s$ is a root of $F$ in the fraction field of the UFD $R$,where $r,s\in R$ without common factor and where $s$ is a nonunit. Then from $F(\alpha)=0$ we get $r^n=-c_{n-1}r^{n-1}s-\cdots-c_1rs^{n-1}-c_0s^n$, saying that $r^n$ and thus $r$ as well are divisible by $s$, contrary to the setup.
In the case at hand, $\mathbb Z_p$ is a Principal Ideal Domain, and thus a Unique Factorization Domain, so that all roots in $\mathbb Q_p$ are automatically in $\mathbb Z_p$.
The polynomial $f(X)=x^3+25 x^2+x-9$ is monic over $\mathbf Z$, by which I mean that it has coefficients in $\mathbf Z$, and the leading coefficient is a $1$. This means that any of its roots will be (by definition) integral over $\mathbf Z$. It is an easy theorem that an element of $\mathbf Q_p$ integral over $\mathbf Z$ in fact lies in $\mathbf Z_p$. Informally, we might put $\overline{\mathbf Z}\cap \mathbf Q_p \subset \mathbf Z_p$, where $\overline{\mathbf Z}$ is the integral closure of $\mathbf Z$ in $\overline{\mathbf Q}$.
This works in much greater generality. If $A$ is any valuation ring (e.g. $\mathbf Z_{(p)}$, $\mathbf Z_p$, and $\mathbf Z_p\cap \overline{\mathbf Q}$) with field of fractions $K$, then $A$ is integrally closed in the sense that if $f\in A[X]$ is a monic polynomial, and $\alpha\in K$ is a root of $f$, then $\alpha\in A$.
Two excellent sources for all of this are: