Quadratic Reciprocity
The Law of Quadratic Reciprocity
If
and
are distinct odd primes then
If
then
is divisible by 4. Sinceis even, the right hand side is one and so is the left hand side therefore. On the other hand if
then
is odd so
We can restate the Law of Quadratic Reciprocity in the form:
Ifandare distinct odd primes then
We can use the Law of Quadratic Reciprocity repeatedly to find if a number is a quadratic residue
For example,
since
since
since 9 is a square number.
The Law of Quadratic Reciprocity can be use to decide if a quadratic equation
has solutions.
has discriminant
since
since
