quartic reciprocity theorem | quartic residue

Gauss stated the reciprocity theorem for the case n = 4 x^4 congruent q (mod p) can be solved using the Gaussian integers as (π/σ)_4 (σ/π)_4 = (-1)^([(N(π) - 1)/4][(N(σ) - 1)/4]). Here, π and σ are distinct Gaussian primes, and N(a + b i) = a^2 + b^2 is the norm. The symbol (α/π) means (α/π)_4 = {1 | if x^4 congruent α (mod π) is solvable -1, i, or - i | otherwise, auto right match where "solvable" means solvable in terms of Gaussian integers.

biquadratic residue | Gaussian integer | Gaussian prime | genus theorem | reciprocity theorem