I want to find all prime and maximal ideals of $\mathbb C[x,y]$ that contain $I=\langle x^2 + 1, y + 3\rangle$.
My approach is that I know that if $f(x)$ is irreducible then $ < f(x) > $ is a prime ideal. So, if I factor $x^2+1$ into $(x+i)(x-i)$ then those are irreducible factors in $C[x,y]$. And $y+3$ is also irreducible over $C[x,y]$. So then the prime ideals that contain $I$ should be: $< x+i, y+3 >$ $< x-i, y+3 >$
Is this true? Can I say that $< x-i, y+3 >$ is prime because $f(x,y) = (x-i) + (y+3)$ is irreducible over $C[x,y]$ or how do I motivate that? I'm kind of unfamiliar with working with ideals in two variabels, and ideals generated by two elements, so I'm not quite sure how I can use the theorems that I know from one-variable.
I also don't know how I can show that these ideals are also maximal. Since $C[x,y]$ is not a PID then I can't draw the conclusion that just because the ideals are prime that they are also maximal.
Also, just for verification that I understood it right. The elements of the ideal $I=< x^2 + 1, y + 3 >$ are all elements of the form $f(x,y) \times (x^2+1) + g(x,y)\times(y+3)$. So therefor it is true that the ideals that I suggested both contain $I$?
Thankful for any help!