Biquadratic Reciprocity Theorem

Gauss stated the reciprocity theorem for the case n=4

 x^4=q (mod p)

can be solved using the Gaussian integers as


Here, pi and sigma are distinct Gaussian primes, and


is the norm. The symbol (alpha/pi) means

 (alpha/pi)_4={1   if x^4=alpha (mod pi) is solvable; -1,i, or -i   otherwise,

where "solvable" means solvable in terms of Gaussian integers.

For a prime number p congruent to 1 (mod 8), 2 is a quartic residue (mod p) if there are integers x,y such that


This is a generalization of the genus theorem. If p is 7 (mod 8), then 2 is always a quartic residue (mod p). In fact, if p=8k+7, then (2^((k+1)))^4 is congruent to 2 (mod p). For example, 2^4 is congruent to 2 (mod 7).

See also

Biquadratic Residue, Gaussian Integer, Gaussian Prime, Genus Theorem, Reciprocity Theorem

Explore with Wolfram|Alpha


Ireland, K. and Rosen, M. "Cubic and Biquadratic Reciprocity." Ch. 9 in A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, pp. 108-137, 1990.

Referenced on Wolfram|Alpha

Biquadratic Reciprocity Theorem

Cite this as:

Weisstein, Eric W. "Biquadratic Reciprocity Theorem." From MathWorld--A Wolfram Web Resource.

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