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we know that the maximal ideals of ${\mathbb Z}[x]$ are of the form $(p, f(x))$ where $p$ is a prime number and $f(x)$ is a polynomial in ${\mathbb Z}[x]$ which is irreducible modulo $p$. Is it true that: the maximal ideals of ${\mathbb Z}[x,y]$ are of the form $(p, f(x,y),g(x,y)$ f(x,y),g(x,y))$ where $p$ is a prime number and $f(x,y), g(x,y)$ are polynomials in ${\mathbb Z}[x,y]$ which are irreducible modulo $p$. |
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Maximal ideals of Z[x,y]we know that the maximal ideals of ${\mathbb Z}[x]$ are of the form $(p, f(x))$ where $p$ is a prime number and $f(x)$ is a polynomial in ${\mathbb Z}[x]$ which is irreducible modulo $p$. Is it true that: the maximal ideals of ${\mathbb Z}[x,y]$ are of the form $(p, f(x,y),g(x,y)$ where $p$ is a prime number and $f(x,y), g(x,y)$ are polynomials in ${\mathbb Z}[x,y]$ which are irreducible modulo $p$.
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