Radical ideal with the empty variety
by Hidenori
Proposition
Show that $\ev{x^2 + 1} \subset \mathbb{R}[x]$ is a radical ideal, but that $V(x^2 + 1)$ is the empty variety.
Solution
$x^2 + 1 \geq 1$ for each $x \in \mathbb{R}$, so $V(x^2 + 1)$ is empty. Since $x^2 + 1$ has no roots in $\mathbb{R}$, $x^2 + 1$ is irreducible in $\mathbb{R}[x]$. By Proposition 9 (P.186, Ideals, Varieties, and Algorithms), $\ev{x^2 + 1}$ is a radical ideal.
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