(a) Let $m \geqslant 2$ be an integer such that $p=4 m-1$ is prime. Suppose that the ideal class group of $L=\mathbb{Q}(\sqrt{-p})$ is trivial. Show that if $n \geqslant 0$ is an integer and $n^{2}+n+m<m^{2}$, then $n^{2}+n+m$ is prime.

(b) Show that the ideal class group of $\mathbb{Q}(\sqrt{-163})$ is trivial.