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In number theory, the law of quadratic reciprocity connects the solvability of two related quadratic equations in modular arithmetic. As a consequence, it allows us to determine the solvability of any quadratic equation in modular arithmetic, even though it does not provide an efficient method for actually finding solutions. To meet Wikipedias quality standards, this article or section may require cleanup. ...
Graph of a quadratic function: y = x2−x−2 = (x+1)(x−2) The x-coordinates of the points where the graph crosses the x-axis, x = −1 and x = 2, are the roots of the quadratic equation: x2−x−2 = 0 In mathematics, a quadratic equation is a polynomial...
Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...
It was conjectured by Euler and Legendre and first satisfactorily proven by Gauss. Gauss called it the 'golden theorem' and was so fond of it that he went on to provide eight separate proofs over his lifetime. Leonhard Euler by Emanuel Handmann. ...
Adrien-Marie Legendre (September 18, 1752 - January 10, 1833) was a French mathematician. ...
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In mathematics, a proof is a demonstration that, assuming certain axioms, some statement is necessarily true. ...
Franz Lemmermeyer's book Reciprocity Laws: From Euler to Eisenstein, published in 2000, collects literature citations for 196 different published proofs for the quadratic reciprocity law. In mathematics, several hundred proofs of the law of quadratic reciprocity have been found. ...
An elementary statement of the theorem Suppose that p and q are two distinct odd prime numbers. The theorem relates the solvability of the equation In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...
 to the solvability of the equation  (see modular arithmetic). There are two cases, depending on whether p and q are congruent to 1 or to 3 (mod 4). Modular arithmetic (sometimes called modulo arithmetic) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value — the modulus. ...
If at least one of p or q is congruent to 1 mod 4 In this case, the theorem says that (A) has a solution if and only if (B) has a solution. That is, either they both have solutions, or they both do not.
For example, if p = 13 and q = 17 (both of which are congruent to 1 mod 4), then (A) has the solution
 and (B) has a solution  On the other hand, if p = 5 and q = 13, then neither (A) nor (B) has a solution (this can be checked by simply listing all of the squares modulo 5 and modulo 13).
The theorem says nothing about the actual solutions themselves, only about whether they exist.
If both p and q are congruent to 3 mod 4 In this case, the theorem says that (A) has a solution if and only if (B) does not have a solution.
For example, if p = 7 and q = 19, then (A) has the solution
 but (B) does not have a solution.
The supplementary theorems There are two extra statements which round out the above laws. Suppose again that p is a prime, not equal to 2. The first says that the equation  has a solution if p is congruent to 1 mod 4, but does not have a solution if it is congruent to 3 mod 4. For example, if p = 29, there is a solution  but for p = 7 there is no solution.
The second says that the equation
 has a solution if and only if p is congruent to 1 or 7 modulo 8, but not if it is congruent to 3 or 5 modulo 8.
Table illustrating quadratic reciprocity The following table illustrates the law of quadratic reciprocity for primes up to 50. In each cell, the first symbol (checkmark or cross) indicates whether p is a square modulo q; the second symbol indicates whether q is a square modulo p. The blue cells are those where either p or q is congruent to 1 modulo 4; the red cells are those where both p and q are congruent to 3 modulo 4. The law of quadratic reciprocity is interpreted as follows: every blue cell contains two identical symbols, and every red cell contains two opposite symbols. | p | | q | | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | 37 | 41 | 43 | 47 | | 3 | | ✗ ✗ | ✓ ✗ | ✗ ✓ | ✓ ✓ | ✗ ✗ | ✓ ✗ | ✗ ✓ | ✗ ✗ | ✓ ✗ | ✓ ✓ | ✗ ✗ | ✓ ✗ | ✗ ✓ | | 5 | ✗ ✗ | | ✗ ✗ | ✓ ✓ | ✗ ✗ | ✗ ✗ | ✓ ✓ | ✗ ✗ | ✓ ✓ | ✓ ✓ | ✗ ✗ | ✓ ✓ | ✗ ✗ | ✗ ✗ | | 7 | ✗ ✓ | ✗ ✗ | | ✓ ✗ | ✗ ✗ | ✗ ✗ | ✗ ✓ | ✓ ✗ | ✓ ✓ | ✗ ✓ | ✓ ✓ | ✗ ✗ | ✓ ✗ | ✗ ✓ | | 11 | ✓ ✗ | ✓ ✓ | ✗ ✓ | | ✗ ✗ | ✗ ✗ | ✗ ✓ | ✓ ✗ | ✗ ✗ | ✓ ✗ | ✓ ✓ | ✗ ✗ | ✗ ✓ | ✓ ✗ | | 13 | ✓ ✓ | ✗ ✗ | ✗ ✗ | ✗ ✗ | | ✓ ✓ | ✗ ✗ | ✓ ✓ | ✓ ✓ | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✓ ✓ | ✗ ✗ | | 17 | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✓ ✓ | | ✓ ✓ | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✓ ✓ | ✓ ✓ | | 19 | ✗ ✓ | ✓ ✓ | ✓ ✗ | ✓ ✗ | ✗ ✗ | ✓ ✓ | | ✓ ✗ | ✗ ✗ | ✗ ✓ | ✗ ✗ | ✗ ✗ | ✓ ✗ | ✓ ✗ | | 23 | ✓ ✗ | ✗ ✗ | ✗ ✓ | ✗ ✓ | ✓ ✓ | ✗ ✗ | ✗ ✓ | | ✓ ✓ | ✓ ✗ | ✗ ✗ | ✓ ✓ | ✗ ✓ | ✓ ✗ | | 29 | ✗ ✗ | ✓ ✓ | ✓ ✓ | ✗ ✗ | ✓ ✓ | ✗ ✗ | ✗ ✗ | ✓ ✓ | | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✗ ✗ | | 31 | ✗ ✓ | ✓ ✓ | ✓ ✗ | ✗ ✓ | ✗ ✗ | ✗ ✗ | ✓ ✗ | ✗ ✓ | ✗ ✗ | | ✗ ✗ | ✓ ✓ | ✗ ✓ | ✓ ✗ | | 37 | ✓ ✓ | ✗ ✗ | ✓ ✓ | ✓ ✓ | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✗ ✗ | | ✓ ✓ | ✗ ✗ | ✓ ✓ | | 41 | ✗ ✗ | ✓ ✓ | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✗ ✗ | ✓ ✓ | ✗ ✗ | ✓ ✓ | ✓ ✓ | | ✓ ✓ | ✗ ✗ | | 43 | ✗ ✓ | ✗ ✗ | ✗ ✓ | ✓ ✗ | ✓ ✓ | ✓ ✓ | ✗ ✓ | ✓ ✗ | ✗ ✗ | ✓ ✗ | ✗ ✗ | ✓ ✓ | | ✓ ✗ | | 47 | ✓ ✗ | ✗ ✗ | ✓ ✗ | ✗ ✓ | ✗ ✗ | ✓ ✓ | ✗ ✓ | ✗ ✓ | ✗ ✗ | ✗ ✓ | ✓ ✓ | ✗ ✗ | ✗ ✓ | |
Statement in terms of the Legendre symbol The theorem can be stated more compactly using the Legendre symbol: Image File history File links Download high resolution version (479x800, 70 KB) Summary This is a scan of page 133 of the first edition of Gauss Disquisitiones Arithmeticae, containing the quadratic reciprocity law. ...
Image File history File links Download high resolution version (479x800, 70 KB) Summary This is a scan of page 133 of the first edition of Gauss Disquisitiones Arithmeticae, containing the quadratic reciprocity law. ...
The Disquisitiones Arithmeticae is a textbook of number theory written by German mathematician Carl Friedrich Gauss and first published in 1801 when Gauss was 24. ...
The Legendre symbol is used by mathematicians in the area of number theory, particularly in the fields of factorization and quadratic residues. ...
 The theorem states that if p and q are two different odd primes, then, using Gauss's original formulation: Odd has several meanings. ...
In mathematics, a prime number (or a prime) is a natural number that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ...
Johann Carl Friedrich Gauss Johann Carl Friedrich Gauss (Gauß) (April 30, 1777 _ February 23, 1855) was a legendary German mathematician, astronomer and physicist with a very wide range of contributions; he is considered to be one of the greatest mathematicians of all time. ...
 if p is of the form 4k + 1 if p is of the form 4k + 3 Which is also equivalent to the very similar form, commonly used today: - if one or both of p and q are of the form 4k + 1
 if both p and q are of the form 4k + 3 Since (p − 1)(q − 1) / 4 is odd if and only if both primes are of the form 4k + 3, we have another commonly-used form:  This is called the main law of quadratic reciprocity, in comparison to the following two supplementary laws (really, theorems): for any odd prime p,  and  The main law of quadratic reciprocity extends to the Jacobi symbol: for positive odd integers m and n which are relatively prime, The Jacobi symbol is used by mathematicians in the area of number theory. ...
In mathematics, the integers a and b are said to be coprime or relatively prime if and only if they have no common factor other than 1 and −1, or equivalently, if their greatest common divisor is 1. ...
 . Notationally, this looks identical to the main law except the parameters are not necessarily prime anymore. The supplementary laws for the Legendre symbol also remain true for the Jacobi symbol, with the odd prime p replaced by an odd positive integer m.
Statement in terms of the Hilbert symbol The quadratic reciprocity law can be formulated in terms of the Hilbert symbol (a,b)v where a and b are any two nonzero rational numbers and v runs over all the non-trivial absolute values of the rationals (the archimedean one and the p-adic absolute values for primes p). The Hilbert symbol (a,b)v is 1 or -1. The Hilbert reciprocity law states that (a,b)v, for fixed a and b and varying v, is 1 for all but finitely many v and the product of (a,b)v over all v is 1. (This formally resembles the residue theorem from complex analysis.)
The proof of Hilbert reciprocity reduces to checking a few special cases, and the non-trivial cases turn out to be equivalent to the main law and the two supplementary laws of quadratic reciprocity for the Legendre symbol. There is no kind of reciprocity in the Hilbert reciprocity law; its name simply indicates the historical source of the result in qudratic reciprocity. Unlike quadratic reciprocity, which requires sign conditions (namely positivity of the primes involved) and a special treatment of the prime 2, the Hilbert reciprocity law treats all absolute values of the rationals on an equal footing. Therefore it is a more natural way of expressing quadratic reciprocity with a view towards generalization: the Hilbert reciprocity law extends with very few changes to all global fields and this extensions can rightly be considered a generalization of quadratic reciprocity to all global fields. The term global field refers to either of the following: a number field, i. ...
Generalizations There are cubic, quartic (biquadratic) and other higher reciprocity laws; but since two of the cube roots of 1 (root of unity) are not real, cubic reciprocity is outside the arithmetic of the rational numbers (and the same applies to higher laws). In mathematics, the nth roots of unity or de Moivre numbers are all the complex numbers which yield 1 when raised to a given power n. ...
See also Gausss lemma in number theory, named for Carl Friedrich Gauss, is involved in some proofs of quadratic reciprocity. ...
The Jacobi symbol is used by mathematicians in the area of number theory. ...
External links - Quadratic Reciprocity Theorem from MathWorld
- A play comparing two proofs of the quadratic reciprocity law
- A complete proof for the Principle of Quadratic Reciprocity
- A proof of this theorem, at PlanetMath
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