I have a question that i can't find an answer anywhere. If I have a monic polynomial $$f=x^n+a_{n-1}x^{n-1}+ \cdots + a_0$$ with $a_i \in \mathbb{Z}$.
If exists a prime $p$ such that $f$ has a factorization ,in $\mathbb{Z_p}$, $\bar{f}=f_{i_1}\cdots f_{i_j}$, where degree($f_{i_k})=i_k$, so if exist a factorization in $\mathbb{Q}$ this one must be $f=g_{i_1}\cdots g_{i_j}$ where degree($g_{i_k})=i_k$.
Example: If i have a monic polynomial of degree $10$ that have in $\mathbb{Z_2}$ a factorization with degrees $(4,3,3)$. If this polynomial were riducible in $\mathbb{Q}$ should have a factorization with degrees $(4,3,3)$?
No.
$X^2+1$ is irreducible in $\Bbb Q[X]$ but $=(X+1)\cdot(X+1)$ in $\Bbb Z_2[X]$. From this we get that the factorization of for example $(X^2+1)(X^9+1)$ in $\Bbb Q[X]$ does not correspond to the factorization in $\Bbb Z_2[X]$ in the way you propose.