# Why $\mathbb Z[X]/I ≃ Z_5[X]/(X + \overline 2)$ and one more

Is the ideal $(X^2 + 1, X + 2)$ prime/maximal in $\mathbb Z[X]?$

$I = (X^2 + 1, X + 2) = (X + 2, 5)$ since $(X + 2)^2 − 4(X + 2) + 5 = X^2 + 1,$ then $\mathbb Z[X]/I ≃ Z_5[X]/(X + \overline 2)$ where $X + \overline 2$ is irreducible in $\mathbb Z_5[X]$ thus the quotient is a field and I is maximal.

I have two questions:

1. Why $\mathbb Z[X]/I ≃ Z_5[X]/(X + \overline 2)?$
2. Why from irreducibility of $X+\overline 2$ follows that quotient field is maximal, what if it wasn't irreducible?

1. We have $\mathbb{Z}[x]/(5) \cong (\mathbb{Z}/5 \mathbb{Z})[x]$, hence $\mathbb{Z}[x]/I\cong \mathbb{Z}/5[x]/(x+2)$.
2. $K[x]/(f)$ is a field if and only if $f$ is irreducible, if and only if $(f)$ is maximal (note that $K[x]$ is a PID for a field $K$).