Two polynomials $f$ and $g$ in $k[x,y]$, $k$ algebraically closed, have the same roots iff they have (up to units) the same factors.

Question

I'm trying to prove this:

Let $f, g \in \textbf{k}[X,Y]$, where $\textbf{k}$ is an algebraically closed field. Then the equations $f(X,Y)=0$ and $g(X,Y)=0$ have the same solutions if and only if $f$ and $g$ have (up to units) the same irreducible factors.

It is difficult? How can I deal with two variables when algebraically closed tells about polynomials in one variable?

Answer 1

Let $Z(f)$ be the zeros of $f$ and $Z(g)$ is defined similarly. Defined $I(X)$ be the ideals of polynomials vanishing on $X$. Then you have $IZ(f)=\sqrt{(f)}$ by Hilbert Nullstellensatz. Hence you have $\sqrt{(f)}=\sqrt{(g)}$. Now you can write $g^n=af$ and $f^m=bg$ for some $a,b\in k[x,y]$ and $n,m\in\mathbb{Z}$. You can see that the irreducible factors of $g$ are also of $f$ and vice versa.

Conversely, it is clear that $f$ and $g$ have the same zeros if they have the same irreducible factors.