The Prime Glossary: Legendre symbol

Legendre symbol
(another Prime Pages' Glossary entries)

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Suppose p is an odd prime and a is any integer. The Legendre symbol (a|p) is defined to be

+1 if a is a quadratic residue (mod p),

-1 if a is a quadratic non-residue (mod p) and,
0 if p divides a

Note: the Legendre symbol is better written vertically:

, but this is difficult and slow on web pages.

Euler showed that (a|p) = a^(p-1)/2 (mod p). Using this we can show the following: Let p and q be odd primes, then

(-1|p) = 1 if p = 1 (mod 4), and (-1|p) = -1 if p = 3 (mod 4);
(a|p) (b|p) = (ab|p);
if a = b (mod p), then (a|p) = (b|p);
(a²|p) = 1 unless p divides a.

For the prime 2 we have

(2|p) = 1 if p = 1 or 7 (mod 8), and
(2|p) = -1 if p = 3 or 5 (mod 8).

Far more difficult to prove is the quadratic reciprocity law:

If p and q are distinct primes, then .

In other words, (p|q) = (q|p), unless p = q = 3 (mod 4), in which case (p|q) = -(q|p).

The Legendre symbol is often evaluated by using the Jacobi symbol.

See Also: JacobiSymbol