Biquadratic Reciprocity Theorem

$$x^4\equiv q\ \left({{\rm mod\ } {p}}\right).$$ (1)

This was solved by Gauß using the Gaussian Integers as

$$\left({\pi\over\sigma}\right)_4\left({\sigma\over\pi}\right)_4=(-1)^{[(N(\pi)-1)/4][(N(\sigma)-1)/4]},$$ (2)

where $\pi$ and $\sigma$ are distinct Gaussian Integer Primes,

$$N(a+bi)=\sqrt{a^2+b^2}$$ (3)

and $N$ is the norm.

$$\left({\alpha\over \pi}\right)_4=\cases{ 1 & if $x^4\equiv \... ...right)$\ is solvable\cr -1, i, {\rm\ or\ } -i & otherwise,\cr}$$ (4)

where solvable means solvable in terms of Gaussian Integers.

See also Reciprocity Theorem