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Gauss published the first and second proofs of the law of quadratic reciprocity on arts 125–146 and 262 of Disquisitiones Arithmeticae in 1801. In number theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime numbers. Due to its subtlety, it ...
Every textbook on elementary number theory (and quite a few on algebraic number theory) has a proof of quadratic reciprocity. Two are especially noteworthy: Lemmermeyer (2000) has many proofs (some in exercises) of both quadratic and higher-power reciprocity laws and a discussion of their history. Its immense bibliography includes literature ...
In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials () with integer coefficients. Recall that first reciprocity law, quadratic reciprocity, determines when an irreducible polynomial f ( x ) = x 2 + a x + b {\displaystyle f(x)=x^{2}+ax+b} splits into linear terms ...
In his second monograph on biquadratic reciprocity, [4]: §§69–71 Gauss used a fourth-power lemma to derive the formula for the biquadratic character of 1 + i in Z[i], the ring of Gaussian integers. Subsequently, Eisenstein used third- and fourth-power versions to prove cubic and quartic reciprocity. [3]: Ch. 8
To test if 2 is a quadratic residue modulo 17, we calculate 2 (17 − 1)/2 = 2 8 ≡ 1 (mod 17), so it is a quadratic residue. To test if 3 is a quadratic residue modulo 17, we calculate 3 (17 − 1)/2 = 3 8 ≡ 16 ≡ −1 (mod 17), so it is not a quadratic residue. Euler's criterion is related to the law of quadratic reciprocity.
The following facts, even the reciprocity laws, are straightforward deductions from the definition of the Jacobi symbol and the corresponding properties of the Legendre symbol. [ 2 ] The Jacobi symbol is defined only when the upper argument ("numerator") is an integer and the lower argument ("denominator") is a positive odd integer.
In number theory, the Legendre symbol is a multiplicative function with values 1, −1, 0 that is a quadratic character modulo of an odd prime number p: its value at a (nonzero) quadratic residue mod p is 1 and at a non-quadratic residue (non-residue) is −1. Its value at zero is 0.
These formulas may be applied repeatedly to get a simple ... all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the ...