Consider the ideal $(f = X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ in the polynomial ring $k[X_0,\ldots, X_n]$. Is this a prime ideal? If so, what is its height? I'm stuck trying to show that $f$ is irreducible.
Is the ideal $(X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ prime?
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commutative-algebra
polynomials
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0Well, I did not mean that the reason for irreducibility is that it has degree one, but that it is obvious how to show it is irreducible because it has degree one :) – 2012-05-16
1 Answers
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Suppose $f$ factored non-trivially. Since $f$ is homogeneous quadratic, we could write $f = L L'$ with $L$, $L'$ homogeneous linear polynomials. What Mariano's argument shows is that any variable $X_i$ cannot occur in both $L$ and $L'$. (Alternatively, if a variable $X_i$ did appear in both $L$ and $L'$, then $f$ would contain a term with $X_i^2$ in it.) Suppose $L$ contains a term of the form $c X_0$ with $c \neq 0$. Then $L'$ must contain the term $\frac{1}{c} X_1$ and cannot contain a term with any other variables, i.e. $L' = \frac{1}{c} X_1$. But the same analysis shows that $L = c X_0$. Conclusion: $f$ factors if and only if $n = 1$.
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0Great, thank you. Do you or someone else have an advice on how to compute the height of this prime, by any chance? – 2019-05-07