This is a follow-up to Projective Spectrum of $K[X,Y]$ .
I see why the given ideals are prime or even maximal, however, I have yet to prove that they in fact make up the entire spectrum of $K[X,Y]$.
Why is it that any polynomial other than the generators mentioned in the previous thread can be factored into degree one polynomials in X and Y?
I know that fixing, say, Y as an element of K will yield a polynomial in $K[X]$ which splits, but this decomposition shouldn't generally be an element of $K[X,Y]$.
Thanks in advance!