Everything about The Legendre Symbol totally explained
The
Legendre symbol or
quadratic character is a function introduced by
Adrien-Marie Legendre in 1798 during his partly successful attempt to prove the
law of quadratic reciprocity.. The symbol has served as the prototype for innumerable higher power residue symbols; other extensions and generalizations include the
Jacobi symbol, the
Kronecker symbol, the
Hilbert symbol and the
Artin symbol. It is one of the earliest examples of a
homomorphism
Definition
The Legendre symbol
The last property is known as the
law of quadratic reciprocity. The properties 4 and 5 are known as the first and second supplements to quadratic reciprocity. They may both be proved from
Gauss's lemma.
Additionally, the Legendre symbol (
a|
p) is a
Dirichlet character (mod
p).
Related functions
The Jacobi symbol is a generalization of the Legendre symbol that allows composite bottom numbers, although the bottom number must still be odd and positive. This generalization provides an efficient way to compute all Legendre symbols.
A further generalization is the Kronecker symbol, which extends the bottom numbers to all integers.Further Information
Get more info on 'Legendre Symbol'.
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