All prime ideals of $\mathbf{R}[x,y]$ and $\mathbf{C}[x, y]$. What are the all the prime ideals of $\mathbf{R}[x,y]$ and $\mathbf{C}[x, y]$? (Also, how do you prove that you've found all of them?) I'm trying to understand what the $\mathbf{R}$-algebraic vs $\mathbf{C}$-algebraic subsets of $\mathbf{C}^2$ are, defined as the zeros (in $\mathbf{C}^2$) of an ideal in $\mathbf{R}[x, y]$ and $\mathbf{C}[x,y]$, respectively. 
Edit: I'm not that familiar with dimension theory, so I'd prefer to see an argument that does not rely on it.

Here's an answer for $\mathbf C[x,y]$:  Hilbert's Nullstellensatz, asserts the maximal ideals  have the form $(x-\alpha,y-\beta)$ for some $\;\alpha, \beta\in \mathbf C$.
On the other hand, $\mathbf C[x,y]$ is a U.F.D. of (Krull) dimension $2$. So the prime ideals of height $1$ are principal, generated by irreducible polynomials.
Add to this list the only ideal of height $0$, and you've finished.
