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...

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

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

{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...

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...

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...

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...

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

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

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reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished...

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...

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

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

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

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...

Arithmeticae (1801). In the introduction to the fifth and sixth proofs of quadratic reciprocity (1818) he said that he was publishing these proofs because their...

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

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...

{\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...

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...

as the Euclid–Euler theorem. Euler also conjectured the law of quadratic reciprocity. The concept is regarded as a fundamental theorem within number...

convenient notational package for several results, including the law of quadratic reciprocity and Euler's criterion. local 1.  A local field in number theory...