23-quadratic-residues - Quadratic residues Michael Freeze MAT 521 UNC Wilmington Fall 2012 1 19 Overview 1 Quadratic Residues 2 The Legendre Symbol 3

23-quadratic-residues - Quadratic residues Michael Freeze...

This preview shows page 1 - 8 out of 19 pages.

Quadratic residues Michael Freeze MAT 521 UNC Wilmington Fall 2012 1 / 19
Image of page 1
Overview 1 Quadratic Residues 2 The Legendre Symbol 3 Gauss’ Lemma 4 Examples 2 / 19
Image of page 2
Outline 1 Quadratic Residues 2 The Legendre Symbol 3 Gauss’ Lemma 4 Examples 3 / 19
Image of page 3
Quadratic Residues Definition Let p be an odd prime and gcd( a , p ) = 1. If the quadratic congruences x 2 a (mod p ) has a solution, then a is said to be a quadratic residue of p . Otherwise, a is called a quadratic nonresidue of p . 4 / 19
Image of page 4
Euler’s Criterion Theorem Let p be an odd prime and gcd( a , p ) = 1. Then a is a quadratic residue of p if and only if a p - 1 2 1 (mod p ). Corollary Let p be an odd prime and gcd( a , p ) = 1. Then a is a quadratic residue or nonresidue of p according to whether a p - 1 2 1 (mod p ) or a p - 1 2 ≡ - 1 (mod p ) . 5 / 19
Image of page 5
Outline 1 Quadratic Residues 2 The Legendre Symbol 3 Gauss’ Lemma 4 Examples 6 / 19
Image of page 6
The Legendre Symbol Definition Let p be an odd prime and let gcd( a , p ) = 1. The Legendre symbol a p is defined by a p = ( 1 if a is a quadratic residue of p - 1 if a is a quadratic nonresidue of p .
Image of page 7
Image of page 8

You've reached the end of your free preview.

Want to read all 19 pages?

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture