Consider the ideal $J =\langle120,x^3-x^2\rangle$ in the ring $\mathbb Z[x]$. How can we decide whether the element $x^3-x+15$ is contained in the radical of $J$ ?
2026-04-04 05:41:34.1775281294
Determining whether an element is contained in the radical of an ideal
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Every polynomial $p \in J$ satisfies $2 \mid p(n)$ for all $n \in \mathbb{Z}$; and therefore, this would also be satisfied for any $p \in \sqrt{J}$. However, $x^3 - x + 15$ does not satisfy this relation.