Let $p$ be a prime such that $p \equiv 5 \pmod {12}$ and let $n$ be a positive integer such that $p > 3^n$. Prove that the ideal class group of $\mathbb{Q}(\sqrt{-p})$ contains an element of order greater than $n$.
I am quite new into class group computations and have absolutely no idea how to start attacking this. Any help appreciated!
The ring of integers is $R=\Bbb Z[\sqrt{-p}]$. In this ring $3$ splits, due to quadratic reciprocity, so $I=\left<3,a+\sqrt{-p}\right>$ is a norm three ideal, because of quadratic reciprocity. Let $1\le k\le n$. Then $I^k$ is not principal. For it has norm $3^k<p$ and the only elements of $R$ with norm $<p$ are ordinary rational integers. So if $I^k$ were principal it would equal $\left<3^{k/2}\right>$. That means $I^2=\left<3\right>$, but that has the prime ideal factorisation $I\overline I$ not $I^2$. So the order of the class of $I$ in the classgroup of $R$ is at least $n+1$.