Can someone help me out with some problem concerning the cardinaliy of ireducible polynomials in $(\mathbb Z/p\mathbb Z)[x]$?
Here first why the problem appeared,
I want to prove that $card(max(p\mathbb Z[x]))=\infty$ where $max(p\mathbb Z[x])$ denotes the set of all maximal ideals containing $p\mathbb Z[x]$ for some prime $p$. So I showed in a first step that $p\mathbb Z[x] + f\mathbb Z[x]$ is an maximal ideal iff $\pi(f)$ is an ireducible polynomial and where $\pi:\mathbb Z[x]\rightarrow(\mathbb Z/p\mathbb Z)[x]$ is the canonical projection.
If now there exists infinitely many ireducible polynomials in $(\mathbb Z/p\mathbb Z)[x]$ i could use the fact that $\pi$ is surjecive to conclude that $card(max(p\mathbb Z[x]))=\infty$ since $p\mathbb Z\subset p\mathbb Z[x] + f\mathbb Z[x] \;\;\forall f\in\mathbb Z[x]$.
But I am not even sure if there are...
Thanks