Proposition

Show that a prime ideal is radical.

Solution

Let $P$ be a prime ideal.

Let $m = 1$. Let $f$ be given such that $f^m \in P$. Then $f \in P$.

Suppose that $f^m \in P \implies f \in P$ for all $f$ for some $m \geq 2$. Let $f$ be given that $f^{m + 1} \in P$. Then $f \in P$ or $f^{m} \in P$. By the inductive hypothesis, $f \in P$.

By mathematical induction, $P$ is radical.