The Law of Quadratic Reciprocity

Ifandare distinct odd primes then

Ifthenis divisible by 4. Sinceis even, the right hand side is one and so is the left hand side therefore. On the other hand ifthenis 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 residueFor example,

since

since

since 9 is a square number.

The Law of Quadratic Reciprocity can be use to decide if a quadratic equationhas solutions.

has discriminant

since

since