Odd-order obstruction for a split prime in an imaginary quadratic field
= Odd-order obstruction for a split prime in an imaginary quadratic field
Let $p\equiv11\pmod{12}$, $K=\mathbb Q(\sqrt{-p})$, and let $\mathfrak p$ be a prime-ideal factor of $(3)$. If $p>3^{n+2}$ and $n$ is odd, then $[\mathfrak p]$ does not have order $n$. Otherwise $\mathfrak p^n=(\alpha)$ would give
$$
4\cdot3^n=a^2+pb^2
$$
for $\alpha=(a+b\sqrt{-p})/2$. The bound forces $b=0$, after which $3^n=N(\alpha)$ is an integer square, impossible for odd $n$.