Quadratic reciprocity
Due to its subtlety, it has many formulations, but the most standard statement is: Law of quadratic reciprocity — Let p and q be distinct odd prime numbers, and define the Legendre symbol as Then This law, together with its supplements, allows the easy calculation of any Legendre symbol, making it possible to determine whether there is an integer solution for any quadratic equation of the form
However, this is a non-constructive result: it gives no help at all for finding a specific solution; for this, other methods are required.
using Euler's criterion one can give an explicit formula for the "square roots" modulo
Generalizing the reciprocity law to higher powers has been a leading problem in mathematics, and has been crucial to the development of much of the machinery of modern algebra, number theory, and algebraic geometry, culminating in Artin reciprocity, class field theory, and the Langlands program.
Quadratic reciprocity arises from certain subtle factorization patterns involving perfect square numbers.
The law of quadratic reciprocity gives a similar characterization of prime divisors of
The quadratic reciprocity law is the statement that certain patterns found in the table are true in general.
The more complicated-looking rules for the quadratic characters of 3 and −5, which depend upon congruences modulo 12 and 20 respectively, are simply the ones for −3 and 5 working with the first supplement.
It is a simple exercise to prove that Legendre's and Gauss's statements are equivalent – it requires no more than the first supplement and the facts about multiplying residues and nonresidues.
Apparently, the shortest known proof yet was published by B. Veklych in the American Mathematical Monthly.
In this article p and q always refer to distinct positive odd primes, and x and y to unspecified integers.
He also claimed to have a proof that if the prime number p ends with 7, (in base 10) and the prime number q ends in 3, and p ≡ q ≡ 3 (mod 4), then Euler conjectured, and Lagrange proved, that[7] Proving these and other statements of Fermat was one of the things that led mathematicians to the reciprocity theorem.
Translated into modern notation, Euler stated [8] that for distinct odd primes p and q: This is equivalent to quadratic reciprocity.
[9] Fermat proved that if p is a prime number and a is an integer, Thus if p does not divide a, using the non-obvious fact (see for example Ireland and Rosen below) that the residues modulo p form a field and therefore in particular the multiplicative group is cyclic, hence there can be at most two solutions to a quadratic equation: Legendre[10] lets a and A represent positive primes ≡ 1 (mod 4) and b and B positive primes ≡ 3 (mod 4), and sets out a table of eight theorems that together are equivalent to quadratic reciprocity: He says that since expressions of the form will come up so often he will abbreviate them as: This is now known as the Legendre symbol, and an equivalent[11][12] definition is used today: for all integers a and all odd primes p He notes that these can be combined: A number of proofs, especially those based on Gauss's Lemma,[13] explicitly calculate this formula.
The sum of these two expressions is Legendre's attempt to prove reciprocity is based on a theorem of his: Example.
positive primes ≡ 3 (mod 4), he breaks it out into the same 8 cases as Legendre: In the next Article he generalizes this to what are basically the rules for the Jacobi symbol (below).
[26] Eisenstein's formula requires relative primality conditions (which are true if the numbers are prime) The quadratic reciprocity law can be formulated in terms of the Hilbert symbol
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.
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.
This proof served as a template for class field theory, which can be viewed as a vast generalization of quadratic reciprocity.
define (ordinary) integers a, b, c, d by the equations, and a function If m = Nμ and n = Nν are both odd, Herglotz proved[34] Also, if Then[35] Let F be a finite field with q = pn elements, where p is an odd prime number and n is positive, and let F[x] be the ring of polynomials in one variable with coefficients in F. If
and f is irreducible, monic, and has positive degree, define the quadratic character for F[x] in the usual manner: If
are monic and have positive degrees,[36] The attempt to generalize quadratic reciprocity for powers higher than the second was one of the main goals that led 19th century mathematicians, including Carl Friedrich Gauss, Peter Gustav Lejeune Dirichlet, Carl Gustav Jakob Jacobi, Gotthold Eisenstein, Richard Dedekind, Ernst Kummer, and David Hilbert to the study of general algebraic number fields and their rings of integers;[37] specifically Kummer invented ideals in order to state and prove higher reciprocity laws.
The ninth in the list of 23 unsolved problems which David Hilbert proposed to the Congress of Mathematicians in 1900 asked for the "Proof of the most general reciprocity law [f]or an arbitrary number field".
[38] Building upon work by Philipp Furtwängler, Teiji Takagi, Helmut Hasse and others, Emil Artin discovered Artin reciprocity in 1923, a general theorem for which all known reciprocity laws are special cases, and proved it in 1927.
Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art.
The two monographs Gauss published on biquadratic reciprocity have consecutively numbered sections: the first contains §§ 1–23 and the second §§ 24–76.
Its immense bibliography includes literature citations for 196 different published proofs for the quadratic reciprocity law.
Kenneth Ireland and Michael Rosen's A Classical Introduction to Modern Number Theory also has many proofs of quadratic reciprocity (and many exercises), and covers the cubic and biquadratic cases as well.
Exercise 13.26 (p. 202) says it all Count the number of proofs to the law of quadratic reciprocity given thus far in this book and devise another one.
