Quadratic reciprocity
Eisenstein integers

Consider the following third root of unity:

The ring of Eisenstein integers is Z. {\displaystyle \mathbb {Z}.} For an Eisenstein prime π, N π ≠ 3, {\displaystyle \pi,\mathrm {N} \pi \neq 3,} and an Eisenstein integer α {\displaystyle \alpha } with gcd (α, π) = 1, {\displaystyle \gcd(\alpha,\pi)=1,} define the quadratic character for Z {\displaystyle \mathbb {Z} } by the formula

Let λ = a + bω and μ = c + dω be distinct Eisenstein primes where a and c are not divisible by 3 and b and d are divisible by 3. Eisenstein proved
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