Odd-order obstruction for a split prime in an imaginary quadratic field

Let $p\equiv 11(\mathrm{mod}12)$, $K=\mathbb{Q}(\sqrt{-p})$, 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 givefor $\alpha =(a+b\sqrt{-p})/2$. The bound forces $b=0$, after which $3^n=N(\alpha )$ is an integer square, impossible for odd $n$.