theory, the law of quadratic reciprocity is a theorem about modular arithmetic that gives conditions for the solvability of quadratic equations modulo prime...

In mathematics, a reciprocity law is a generalization of the law of quadratic reciprocity to arbitrary monic irreducible polynomials f ( x ) {\displaystyle...

In number theory, the law of quadratic reciprocity, like the Pythagorean theorem, has lent itself to an unusually large number of proofs. Several hundred...

1797 or 1798 in the course of his attempts at proving the law of quadratic reciprocity. Generalizations of the symbol include the Jacobi symbol and Dirichlet...

introduced the square bracket notation [x] in his third proof of quadratic reciprocity (1808). This remained the standard in mathematics until Kenneth...

a quadratic residue or nonresidue modulo n {\displaystyle n} . The Kronecker symbol also satisfies the following versions of quadratic reciprocity law...

nonresidue. The first supplement to the law of quadratic reciprocity is that if p ≡ 1 (mod 4) then −1 is a quadratic residue modulo p, and if p ≡ 3 (mod 4) then...

density theorem. The law of quadratic reciprocity implies that the splitting behaviour of a prime p {\displaystyle p} in a quadratic field depends only on p...

{q-1}{2}}}.} A reciprocity law is a generalization of the law of quadratic reciprocity. There are several different ways to express reciprocity laws. The early...

seems to have come from the study of higher reciprocity laws, that is, generalizations of quadratic reciprocity. Number fields are often studied as extensions...

and applied them to quadratic, cubic, and biquadratic reciprocity laws. For an odd prime number p and an integer a, the quadratic Gauss sum g(a; p) is...

the law of quadratic reciprocity and the Fermat polygonal number theorem. He also contributed to the theory of binary and ternary quadratic forms, the...

term "reciprocity law" refers to a long line of more concrete number theoretic statements which it generalized, from the quadratic reciprocity law and...

prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the existence...

ISBN 9780521585330 Russinoff, David M. (1992), "A Mechanical Proof of Quadratic Reciprocity", J. Autom. Reason., 8 (1): 3–21, doi:10.1007/BF00263446, S2CID 14824949...

residue r is a quadratic residue (mod q) if and only if it is a biquadratic residue (mod q). Indeed, the first supplement of quadratic reciprocity states that...

all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished...

Modular_arithmetic#Residue_class Quadratic_residue#Prime_power_modulus Artin symbol Gauss's lemma Quadratic reciprocity deals with squares; higher refers...

point of the program was Emil Artin's reciprocity law, which generalizes quadratic reciprocity. The Artin reciprocity law applies to a Galois extension of...

Carl Friedrich Gauss referred to the law of quadratic reciprocity as the "fundamental theorem" of quadratic residues. There are also a number of "fundamental...

characteristics Helmholtz reciprocity, linear propagation. Reciprocity law (law of reciprocity) in mathematics, including Quadratic reciprocity, a fundamental result...

in time O((log n)²) using Jacobi's generalization of the law of quadratic reciprocity. Given an odd number n one can contemplate whether or not the congruence...

and martingales Quadratic reciprocity, a theorem from number theory Quadratic residue, an integer that is a square modulo n Quadratic sieve, a modern...

Reciprocity theorem may refer to: Quadratic reciprocity, a theorem about modular arithmetic Cubic reciprocity Quartic reciprocity Artin reciprocity Weil...

{\displaystyle -1} for the remaining. It is easy to compute using the law of quadratic reciprocity in a manner akin to the Euclidean algorithm; see Legendre symbol...

Gaussian integer, Gaussian rational Quadratic field Cyclotomic field Cubic field Biquadratic field Quadratic reciprocity Ideal class group Dirichlet's unit...

theorem that has been proved in many different ways is the theorem of quadratic reciprocity. In fact, Carl Friedrich Gauss alone had eight different proofs...

number theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it...

theory Eisenstein's reciprocity law is a reciprocity law that extends the law of quadratic reciprocity and the cubic reciprocity law to residues of higher...

quadratic residue. Although it is not useful computationally, it has theoretical significance, being involved in some proofs of quadratic reciprocity...