Reciprocity theorems

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Reciprocity theorem

If there exists a rational integer such that, when , , and are positive integers,then is the -adic residue of , i.e., is an -adic residue of iff is solvable for . Reciprocity theorems relate statements of the form " is an -adic residue of " with reciprocal statements of the form " is an -adic residue of ."The first case to be considered was (the quadratic reciprocity theorem), of which Gauss gave the first correct proof. Gauss also solved the case (cubic reciprocity theorem) using integers of the form , where is a root of and , are rational integers. Gauss stated the case (biquadratic reciprocity theorem) using the Gaussian integers.Proof of -adic reciprocity for prime was given by Eisenstein in 1844-50 and by Kummer in 1850-61. In the 1920s, Artin formulated Artin's reciprocity theorem, a general reciprocity law for all orders...

Quadratic residue

If there is an integer such that(1)i.e., the congruence (1) has a solution, then is said to be a quadratic residue (mod ). Note that the trivial case is generally excluded from lists of quadratic residues (e.g., Hardy and Wright 1979, p. 67) so that the number of quadratic residues (mod ) is taken to be one less than the number of squares (mod ). However, other sources include 0 as a quadratic residue.If the congruence does not have a solution, then is said to be a quadratic nonresidue (mod ). Hardy and Wright (1979, pp. 67-68) use the shorthand notations and , to indicated that is a quadratic residue or nonresidue, respectively.In practice, it suffices to restrict the range to , where is the floor function, because of the symmetry .For example, , so 6 is a quadratic residue (mod 10). The entire set of quadratic residues (mod 10) are given by 1, 4, 5, 6, and 9, since (2)(3)(4)making the numbers 2, 3, 7, and 8 the quadratic nonresidues (mod 10).A list..

Quadratic reciprocity theorem

If and are distinct odd primes, then the quadratic reciprocity theorem states that the congruences(1)are both solvable or both unsolvable unless both and leave the remainder 3 when divided by 4 (in which case one of the congruences is solvable and the other is not). Written symbolically,(2)where(3)is known as a Legendre symbol.Gauss called this result the "aureum theorema" (golden theorem).Euler stated the theorem in 1783 without proof. Legendre was the first to publish a proof, but it was fallacious. In 1796, Gauss became the first to publish a correct proof (Nagell 1951, p. 144). The quadratic reciprocity theorem was Gauss's favorite theorem from number theory, and he devised no fewer than eight different proofs of it over his lifetime.The genus theorem states that the Diophantineequation(4)can be solved for a prime iff or ...

Quadratic nonresidue

If there is no integer such thati.e., if the congruence (35) has no solution, then is said to be a quadratic nonresidue (mod ). If the congruence (35) does have a solution, then is said to be a quadratic residue (mod ).In practice, it suffices to restrict the range to , where is the floor function, because of the symmetry .The following table summarizes the quadratic nonresidues for small (OEIS A105640).quadratic nonresidues1(none)2(none)3242, 352, 362, 573, 5, 682, 3, 5, 6, 792, 3, 5, 6, 8102, 3, 7, 8112, 6, 7, 8, 10122, 3, 5, 6, 7, 8, 10, 11132, 5, 6, 7, 8, 11143, 5, 6, 10, 12, 13152, 3, 5, 7, 8, 11, 12, 13, 14162, 3, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15173, 5, 6, 7, 10, 11, 12, 14182, 3, 5, 6, 8, 11, 12, 14, 15, 17192, 3, 8, 10, 12, 13, 14, 15, 18202, 3, 6, 7, 8, 10, 11, 12, 13, 14, 15, 17, 18, 19The numbers of quadratic nonresidues (mod ) for , 2, ... are 0, 0, 1, 2, 2, 2, 3, 5, 5, 4, 5, 8, 6, 6, ... (OEIS A095972).The smallest quadratic nonresidues for , 4, ... are 2, 2, 2, 2, 3, 2, 2, 2, 2, 2,..

Genus theorem

The Diophantine equationcan be solved for a prime iff or . The representation is unique except for changes of sign or rearrangements of and . This theorem is intimately connected with the quadratic reciprocity theorem, and generalizes to the biquadratic reciprocity theorem.

Gauss's lemma

Let the multiples , , ..., of an integer such that be taken. If there are an even number of least positive residues mod of these numbers , then is a quadratic residue of . If is odd, is a quadratic nonresidue. Gauss's lemma can therefore be stated as , where is the Legendre symbol. It was proved by Gauss as a step along the way to the quadratic reciprocity theorem (Nagell 1951).The following result is known as Euclid's lemma, but is incorrectly termed "Gauss's Lemma" by Séroul (2000, p. 10). Euclid's lemma states that for any two integers and , suppose . Then if is relatively prime to , then divides .

Euler's quadratic residue theorem

A number that possesses no common divisor with a prime number is either a quadratic residue or nonresidue of , depending whether is congruent mod to .

Cubic residue

If there is an integer such thatthen is said to be a cubic residue (mod ). If not, is said to be a cubic nonresidue (mod ).

Cubic reciprocity theorem

A reciprocity theorem for the case solved by Gauss using "integers" of the form , when is a root of (i.e., equals or ) and , are integers.

Biquadratic residue

If there is an integer such thatthen is said to be a biquadratic residue (mod ). If not, is said to be a biquadratic nonresidue (mod ).

Biquadratic reciprocity theorem

Gauss stated the reciprocity theorem for the case (1)can be solved using the Gaussian integers as(2)Here, and are distinct Gaussian primes, and(3)is the norm. The symbol means(4)where "solvable" means solvable in terms of Gaussianintegers.For a prime number congruent to 1 (mod 8), 2 is a quartic residue (mod ) if there are integers such that(5)This is a generalization of the genus theorem. If is 7 (mod 8), then 2 is always a quartic residue (mod ). In fact, if , then is congruent to 2 (mod ). For example, is congruent to 2 (mod 7).

Associate

Let be an odd prime, be a positive number such that (i.e., does not divide ), and let be one of the numbers 1, 2, 3, ..., . Then there is a unique , called the associate of , such thatwith (Hardy and Wright 1979, p. 67). If , then is called a quadratic residue of .

Artin's reciprocity theorem

A general reciprocity theorem for all orders which covered all other known reciprocity theorems when proved by E. Artin in 1927. If is a number field and a finite integral extension, then there is a surjection from the group of fractional ideals prime to the discriminant, given by the Artin symbol. For some cycle , the kernel of this surjection contains each principal fractional ideal generated by an element congruent to 1 mod .

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