Let $p\equiv 2\pmod 3$ be a prime number such that $p>3^m$ for some $m\in\mathbb N$. Prove that the class number of $K=\mathbb{Q}(\sqrt{-p})$ is bounded below by $m$.
I thought using Minkowski bound given by $M\ge\frac{2}{\pi}\sqrt{p}>\frac{2}{\pi}3^{\frac m2}.$ That does mean I can look at prime ideals diving $2\mathcal{O}_K, 3\mathcal{O}_K$ but how does that help? How can I deduce the class number from this bound?
Prove first that there is a prime ideal $\frak p$ of norm $3$, then prove that none of $\frak p$, $\frak p^2,\ldots,\frak p^m$ are principal.
The Minkowski bound will only give upper bounds on the class number.