# An irreducible polynomial over $\mathbb{R}[x,y]$

Let $(x_i,y_i)$ be a finite subset of points of $\mathbb{R}^2$. Find an irreducible polynomials $f(x,y)$ over $\mathbb{R}[x,y]$ such that vanish only in that points.

EDITED: Where $\mathbb{R}$ denoted the Real numbers.

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Is $R$ just any ring or is it the reals $\mathbb R$ ? – lhf Apr 12 '12 at 16:23
Hint: The polynomial $f_i(x,y)=(x-x_i)^2+(y-y_i)^2$ vanishes at $(x_i,y_i)$ but not at any other point. How can you put together a collection of functions $\{f_i\}$ to get a single function which vanishes wherever one of the $f_i$ vanishes?