I had to solve many exercises to take my algebra exam lately and lots of them are on polynomials, whereas I am asked to look for roots and verify if the given polynomials are reducible in a specific $F[x]$ field or not.
I have noticed something while doing these exercises and I don't really know if my intuition is supported by some actual theorem (and that is why I am asking for your kind help), so I am wondering:
say $p(x)$ is a polynomial over $\mathbb Z_2$, is it possible that if $p(x)$ is irreducible in $\mathbb Z_2$ then it is irreducible in every $\mathbb Z_n \text { } (\forall n > 2)$?
and generally speaking is is true that if a polynomial $p(x)$ is irreducible in a certain $\mathbb Z_k \Rightarrow p(x)$ is irreducible in $\mathbb Z_j$ $(\forall j > k)$?