Is the ideal $(X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ prime?

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Is the ideal $(X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ prime?

Consider the excellent $(f = X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ in the polynomial ring $k[X_0,\ldots, X_n]$. Is this a prime perfect? If so, what is its elevation? I'm stuck attempting to show that $f$ is irreducible.

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math commutative-algebra polynomials

jonny 2022-07-25 20:47:17

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Is the ideal $(X_0X_1+X_2X_3+\ldots+X_{n-1}X_n)$ prime?