Let R denote the real numbers, let´s take a finite number of points in $R^2$ 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 exist an element on this ideal that is an irreducible polynomial. Clearly I can suppose that all the finite generators, are not irreducible , otherwise it´s done. How using this I can find such polynomial?