Editing Legendre symbol (section)

== Legendre symbol and quadratic reciprocity ==
Let ''p'' and ''q'' be distinct odd primes. Using the Legendre symbol, the [[quadratic reciprocity]] law can be stated concisely:

: <math>\left(\frac{q}{p}\right)\left(\frac{p}{q}\right) = (-1)^{\tfrac{p-1}{2}\cdot\tfrac{q-1}{2}}.</math>

Many [[proofs of quadratic reciprocity]] are based on Euler's criterion

:<math>\left(\frac{a}{p}\right) \equiv a^{\tfrac{p-1}{2}} \pmod p.</math>

In addition, several alternative expressions for the Legendre symbol were devised in order to produce various proofs of the quadratic reciprocity law.

* Gauss introduced the [[quadratic Gauss sum]] and used the formula

::<math>\sum_{k=0}^{p-1}\zeta^{ak^2}=\left(\frac{a}{p}\right)\sum_{k=0}^{p-1}\zeta^{k^2},\qquad \zeta = e^{\frac{2\pi i}{p}}</math>

:in his fourth<ref>Gauss, "Summierung gewisser Reihen von besonderer Art" (1811), reprinted in ''Untersuchungen ...'' pp. 463–495</ref> and sixth<ref>Gauss, "Neue Beweise und Erweiterungen des Fundamentalsatzes in der Lehre von den quadratischen Resten" (1818) reprinted in ''Untersuchungen ...'' pp. 501–505</ref> proofs of quadratic reciprocity.

* [[Leopold Kronecker|Kronecker's]] proof<ref>Lemmermeyer, ex. p. 31, 1.34</ref>  first establishes that

::<math>\left(\frac{p}{q}\right) =\sgn\left(\prod_{i=1}^{\frac{q-1}{2}}\prod_{k=1}^{\frac{p-1}{2}}\left(\frac{k}{p}-\frac{i}{q}\right)\right).</math>

: Reversing the roles of ''p'' and ''q'', he obtains the relation between ({{Sfrac|''p''|''q''}}) and ({{Sfrac|''q''|''p''}}).

* One of [[Gotthold Eisenstein| Eisenstein]]'s proofs<ref>Lemmermeyer, pp. 236 ff.</ref> begins by showing that

::<math>\left(\frac{q}{p}\right) =\prod_{n=1}^{\frac{p-1}{2}} \frac{\sin\left(\frac{2\pi qn}{p}\right)}{\sin\left(\frac{2\pi n}{p}\right)}.</math>

: Using certain [[elliptic function]]s instead of the [[sine function]], [[Eisenstein reciprocity|Eisenstein]] was able to prove [[cubic reciprocity|cubic]] and [[quartic reciprocity]] as well.