For prime numbers $p$ such that $p \equiv 11$, $13$, $17$, $19 \text{ mod }20$, does the prime ideal $(p)$ of $\mathbb{Z}[\sqrt{-5}]$ split completely in the extension of $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$? I would like to avoid citing any overpowered result from class field theory to see an answer to this question, if possible.
Thoughts. I know that if $p \equiv 11$, $19 \text{ mod }20$, $p$ splits completely in $\mathbb{Q}(\sqrt{5})/\mathbb{Q}$, and I also know that if $p \equiv 13$, $17 \text{ mod }20$, $p$ splits completely in $\mathbb{Q}(i)/\mathbb{Q}$.
Yes this is true, and you already have most of the ideas.
For example, when $p\equiv 11,19\pmod{20}$, since $\mathbb Q(\sqrt {-5},i)$ is the compositum of $\mathbb Q(\sqrt5)$ and $\mathbb Q(\sqrt{-5})$ and $p$ splits in $\mathbb Q(\sqrt 5)$, $p$ must split in $\mathbb Q(\sqrt {-5},i)/\mathbb Q$.
But $p$ is inert in $\mathbb Q(\sqrt{-5})$, so the splitting must take place in the extension $\mathbb Q(\sqrt {-5},i)/\mathbb Q(\sqrt {-5})$.