Example of polynomial in two variables

Question

Can you please give me an example of a polynomial $F \in K[X,Y]$ such that $V(F)$ is finite?

I found in Fulton the following proposition:

If F is an irreducible polynomial in $K[X,Y]$ such that $V(F)$ is infinite, then $I(V(F))=(F)$, and $V(F)$ is irreducible.

Is the condition $V(F)$ is infinite necessarily?

Thank you!

Answer 1

Take $K = \mathbb{R}$ and $F(X,Y) = X^2 + Y^2$.

Answer 2

Here's a very easy way: take $K$ to be a finite field and $F$ to be any polynomial in $K[X,Y]$.