Let $M$ be a maximal ideal of the ring $\mathbb{Z}[x]$. I want to show that $M$ must contain some prime number $p$.
My attempt
Assume not. We know $M$ can't contain $1$. Let $q$ be the smallest positive integer in $M$. Then $q$ must be composite, say $p|q$ for some prime $p$. Define $M_p:=${$α+pβ : \alpha\in M$ and $\beta\in\mathbb{Z}[x]$}. It is easy to see that $M_p$ is an ideal of $\mathbb{Z}[x]$.
Now, if I can show that $1\notin M_p,$ this should imply a contradiction. But how can I show that $1\notin M_p$? I am struggling with this.