Chapter 3 Definitions

Question 1

A Quadratic Residue

Answer 1

Let p be an odd prime and a an integer not divisible by p.

We say that a is a quadratic residue of p when

x² ≡ a (mod p)

has at least one solution and a quadratic non-residue other wise.

Question 2

The Legendre Symbol

Answer 2

The Legendre Symbol (a/p)

is defined by:

• (a/p) =*
• +1 if p∤a and x² ≡ a (mod p) has a solution,*
• -1 if p∤a and x2 ≡ a (mod p) does not have a solution,*
• 0 if p|a*

Question 3

The number of incongruent solutions to x² ≡ a (mod p)

Answer 3

is 1+(a/p)