Why is every prime $3,7 (\bmod 20)$ of the form
$$ 2 x^2 + 2 xy + 3 y^2 $$
I do not think that form is the norm of an abelian ring ?
How to prove this ?
2026-04-03 09:28:12.1775208492
Primes of the form $ 2 x^2 + 2 xy + 3 y^2 $
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These are the norms of the non-principal ideals of $\Bbb Z[\sqrt{-5}]$. The ideals of prime norm in this ring have norms $2$, $5$ and $p$ with $p\equiv1,3,7,9\pmod{20}$. The principal ideals cannot have norms $\equiv3,7$ and the non-principal ones cannot have norms $\equiv1,9$.