Primes inert in quadratic field of class number one

Solution 1:

Let $p$ a prime. If ${\frak P}$ is a prime ideal of $O_K$ above $p$, then the assumption on the class number implies that ${\frak P}=\langle \alpha\rangle$ for some $\alpha\in O_K$. Let $\alpha=a+b\theta$, where $\theta$ is either $\sqrt{-d}$ or $(1+\sqrt{-d})/2$ depending. Show that if $\alpha$ is not a rational integer, then $N(\alpha)\ge(1+|d_K|)/4$.

If $p$ splits, then we must have $N(\alpha)=p$. This is clearly impossible for a rational integer $\alpha$. So the above result shows that $p\ge(1+|d_K|)/4$.

Solution 2:

Well, according to Stark-Heegner, there are only 9 such fields, corresponding to the values of $-d$ $$−1, −2, −3, −7, −11, −19, −43, −67, −163.$$ Thus, the statement can be verified by a finite computation (left to the reader ;)).