I know that the maximal ideals (via the weak nullstellensatz) are precisely 
$(x - a, y - b)$ for $a, b \in \mathbf{C}$. My question is how to show this using the following fact. We can show else that for any ring $k[x,y]$ with $k$ a field, maximal ideals take the form $(p, g)$ with $p$ a prime in $k[x]$, and $g \in k[x,y]$ such that $\overline{g} \in (k[x]/(p))[y]$ is irreducible. Thus a maximal ideal looks like $(x - a, g(x, y))$ with some condition on $g$. 

**Question 1:** How do you show that these ideals can be written with $g$ taken to be $y- b$ with $b \in \mathbf{C}$? 

**Question 2:** How do you do this kind of computation if instead of $k = \mathbf{C}$, you took $k = \mathbf{R}$? You can't use the nullstellensatz, but you can to use this kind of computation. I guess it boils down to showing (A.) what the prime ideals of $\mathbf{R}[x]$ are, and (B.) what the elements $g$ are that satisfy the condition described above. Can you explain how to do this computation in the case of $\mathbf{R}$?