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

Famous quotes containing the words symbol and/or reciprocity:

    In a symbol there is concealment and yet revelation: here therefore, by silence and by speech acting together, comes a double significance.... In the symbol proper, what we can call a symbol, there is ever, more or less distinctly and directly, some embodiment and revelation of the Infinite; the Infinite is made to blend itself with the Finite, to stand visible, and as it were, attainable there. By symbols, accordingly, is man guided and commanded, made happy, made wretched.
    Thomas Carlyle (1795–1881)

    Between women love is contemplative; caresses are intended less to gain possession of the other than gradually to re-create the self through her; separateness is abolished, there is no struggle, no victory, no defeat; in exact reciprocity each is at once subject and object, sovereign and slave; duality become mutuality.
    Simone De Beauvoir (1908–1986)