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