Are $(X+1,X), (X^2+4,5)$ and $(X^2+1,X+2)$ maximal or prime?

How do I know if $(X+1,X), (X^2+4,5)$ and $(X^2+1,X+2)$ are maximal or prime in $\mathbb{Z}[X]$?

I know the definitions of maximal and prime ideal, but I don't know how to do this exercise? Any hint?

If I'm not wrong, $(X+1,X)=(1,X)$ and $(X^2+1,X+2)=(5,X+2)$ but I don't know if this will help me...

Solution 1:

I'll write a solution for the middle one. Similar games apply to the other ones---investigate whether the quotients are integral domains, fields, or neither.

We have $$ \mathbb Z[X] / (X^2 + 4, 5) \simeq (\mathbb Z / 5 \mathbb Z)[X] / (X^2 + 4), $$ so we are considering $X^2 + 4$ as a polynomial in the ring of polynomials with coefficients modulo $5$. Note that modulo $5$, the polynomial $$ X^2 + 4 = X^2 - 1 = (X - 1)(X + 1) = (X + 4)(X + 1) $$ factors, so the polynomial is certainly not irreducible, and hence the ideal is definitely not maximal. But it is also not prime, for the same factorisation tells us that in the quotient $\mathbb Z[X] / (X^2 + 4, 5)$ we have zero divisors!