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

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

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?

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?

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Daniel
Daniel
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irreducible elements in a ideal of $R[x_1,x_2]$

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?