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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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Welcome to math.SE: since you are fairly new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are so far; this will prevent people from telling you things you already know, and help them write their answers at an appropriate level. If this is homework, please add the [homework] tag; people will still help, so don't worry. Also, many find the use of imperative ("Find", "Show") to be rude when asking for help; please consider rewriting your post. –  Arturo Magidin Apr 12 '12 at 3:16
    
Arkj has just quoted some exercise. I have no idea why this should be rude ... by the way Arkj has problems with finding an irreducible polynomial, which seems to be much harder than finding an arbitrary polynomial with the desired property. –  Martin Brandenburg Apr 12 '12 at 7:54
    
Is $R$ just any ring or is it the reals $\mathbb R$ ? –  lhf Apr 12 '12 at 16:23

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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?

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@Arturo Magidin Sorry for use that words, but I don´t know to much english, so I don´t know what words sounds rude ,etc. I tried a lot of things, but the property of being irreducible is the main problem :/! –  Arkj Apr 12 '12 at 4:17

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