which primes $\mathfrak{p}$ can be written as sums of two squares in $\mathbb{Q}(\sqrt[3]{2})$? I am asking to solve an equation:
$$ \mathfrak{p} = \big(a_1+a_2\sqrt[3]{2}+a_3\sqrt[3]{4}\big)^2 + \big(b_1+b_2\sqrt[3]{2}+b_3\sqrt[3]{4}\big)^2 $$
I am still figuring out the code in Sage or PARI/GP. Perhaps there's a lattice or ideal $I \subseteq \mathbb{Z}[\sqrt[3]{2}]$ such that this sum-of-squares equation has a solution iff $\mathfrak{p} \in I$. Have no idea really
Why do I have hope for asking this question? Here's an vaguely related example from an article of Jared Weinstein:
$x^4 - 2$ splits modulo $p$ iff $p = a^2 + 64b^2$ for integers $a$ and $b$.
This becomes a statement about algebraic number fields:
Let $K = \mathbb{Q}(i)$ and $L = K(\sqrt[4]{2})$ .
This means $[L:K]=4$ and $[K:\mathbb{Q}]= 2$ so that $[L:\mathbb{Q}]=[L:K][K:\mathbb{Q}]=4 \times 2 = 8$.
Let $p = a^2 + 64b^2$ and $a \equiv 1 \pmod 8$, then $p = \pi \overline{\pi}$ where $\pi \in \mathbb{Z}[i]$ is a Gaussian prime and $\pi \equiv 1 \pmod{ 8 \mathbb{Z}[i]}$ and so $p$ splits in $L$.