Let $\mathbf R$ denote the real numbers, let's take a finite number of points in $\mathbf R^2$ and let's take the ideal $I$ of all the polynomials that vanish on this points. Using the Hilbert basis theorem we know that $I$ is finitely generated. I want to know if there exists an element in this ideal that is an irreducible polynomial. 

Clearly I can suppose that all the finite generators are not irreducible, otherwise it's done. Using this, how can I find such a polynomial?