Gauss stated the reciprocity theorem for the case
(1)

can be solved using the Gaussian integers as
(2)

Here, and are distinct Gaussian primes, and
(3)

is the norm. The symbol means
(4)

where "solvable" means solvable in terms of Gaussian integers.
For a prime number congruent to 1 (mod 8), 2 is a quartic residue (mod ) if there are integers such that
(5)

This is a generalization of the genus theorem. If is 7 (mod 8), then 2 is always a quartic residue (mod ). In fact, if , then is congruent to 2 (mod ). For example, is congruent to 2 (mod 7).