Problem 5

Let $I=\{f(x)\in\mathbb{Z}[x]\mid f(-2)=0\}$ be a subset of $\mathbb{Z}[x]$. Consider the homomorphism $\phi:\mathbb{Z}[x]\to\mathbb{Z}$ where $\phi(f(x))=f(-2)$ for all $f(x)\in\mathbb{Z}[x]$.

1. Show that $I$ is an ideal of $\mathbb{Z}[x]$.

2. Hence determine if the ideal $I$ is prime, maximal, or both.