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Biquadratic Reciprocity Theorem

Gauss stated the reciprocity theorem for the case n=4

x^4=q (mod p)
(1)

can be solved using the Gaussian integers as

(pi/sigma)_4(sigma/pi)_4=(-1)^([(N(pi)-1)/4][(N(sigma)-1)/4]).
(2)

Here, pi and sigma are distinct Gaussian primes, and

N(a+bi)=a^2+b^2
(3)

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,
(4)

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

x^2+64y^2=p.
(5)

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

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References

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. https://mathworld.wolfram.com/BiquadraticReciprocityTheorem.html

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