Characterise ideals in a ring I am trying to determine whether the following ideals of $\mathbb{Z}[X]$ are prime or maximal ideals:


*

*$(X^2 - 3)$,

*$(5,X^2 + 3)$. 
I am trying to do this by establishing whether $\mathbb{Z}[X]/I$ is a field or domain. For (2), I have the following attempt so far: 
$$
\mathbb{Z}[X]/(5,X^2 + 3) \cong (\mathbb{Z}[X]/(5)) / ((5,X^2 + 3)/(5)) \cong (\mathbb{Z}/5 \mathbb{Z})[X] / (\overline{X}^2 + \overline{3})
$$
I don't know how to proceed from here or if I'm on the right track. 
For (1), I wanted to show that $\phi: \mathbb{Z}[X] \to \mathbb{R} : f \mapsto f(\sqrt{3})$ is a homomorphism with kernel $(X^2 - 3)$, but I am not sure if it is surjective onto $\mathbb{R}$. 
Anyone can help me further on this?
 A: Your ideas are good. For $(X^2-3)$, you will want to consider the map $f:Z[x]\to \mathbb{Z}[\sqrt{3}]$ to get an isomorphism with $\mathbb{Z}[x]/(x^2-3)$, but even if you consider the map to $\mathbb{R}$ that is ok, since the image is a subring (the map is not surjective) of $\mathbb{R}$ which is necessarily an integral domain. 
For the next part, note that $\overline{X}^2+\overline{3}$ is irreducible as a polynomial in $\mathbb{F}_5$, since $\overline{2}$ is not a quadratic residue in that field. Can you go from there?
A: For $2$,  you're on the right track. Just observe that $\mathbf Z/5\mathbf Z$ is a field, hence the polynomial ring $\mathbf Z/5\mathbf Z[X]$ is a P.I.D. so all you have to prove is that $X^2+3$ is irreducible in  $\mathbf Z/5\mathbf Z[X]$. As it is a quadratic polynomial, it amounts to proving it has no roots mod. $5$, i.e. proving $-3$ is not a quadratic residue mod.5. Now the squares mod.5 are $0,1$ and $-1$.
For $1$, sme method, but simpler: if $X^2-3$ were not irreducible it would have integer roots, which is not the case. Now in a P.I.D. (more generally in a U.F.D.), irreducible elements generate a prime ideal. This ideal is not maximal   because, for instance it is (strictly) contained in the ideal 
$(X^2-3,5)$ – which, incidentally, is maximal.
