Legendre Symbol - Legendre Symbol and Quadratic Reciprocity

Legendre Symbol and Quadratic Reciprocity

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

\left(\frac{q}{p}\right) = \left(\frac{p}{q}\right)(-1)^{\tfrac{p-1}{2}\tfrac{q-1}{2}}.

Many proofs of quadratic reciprocity are based on Legendre's formula

\left(\frac{a}{p}\right) \equiv a^{\tfrac{p-1}{2}}\pmod p.

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
\sum_{k=0}^{p-1}\zeta^{ak^2}= \left(\frac{a}{p}\right)\sum_{k=0}^{p-1}\zeta^{k^2}, \quad \zeta = e^{2\pi i/p}
in his fourth and sixth proofs of quadratic reciprocity.
  • Kronecker's proof first establishes that
\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).
Reversing the roles of p and q, he obtains the relation between and
  • One of Eisenstein's proofs begins by showing that
\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)}.
Using certain elliptic functions instead of the sine function, Eisenstein was able to prove cubic and quartic reciprocity as well.

Read more about this topic:  Legendre Symbol

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