Quadratic Residues

Question 1

quadratic residue mod m

Answer 1

gcd(a,m)=1 and a=x^2 mod m

Question 2

quadratic non residue

Answer 2

gcd(a,m)=1 and not a quadratic residue

Question 3

Legendre symbol

Answer 3

p odd prime
LS= 1 if a is a quadratic residue
-1 if a is a quadratic non residue
0 if p divides a

Question 4

Euler’s criterion

Answer 4

p odd prime
(a/p) = a^(p-1)/2 mod p

Question 5

Basic properties

Answer 5

-respects congruence
-if a is a square then LS=1
-multiplicative

Question 6

-1 case

Answer 6

1 if p=1 mod 4
-1 if p=3 mod4

Question 7

2 case

Answer 7

1 if p=+/-1 mod 8
-1 if p=+/-3 mod 8

Question 8

Law of Quadratic Reciprocity

Answer 8

p,q distinct odd primes
invert: - if both are 3 mod 4
+ otherwise

Question 9

Jacobi symbol

Answer 9

odd numbers- not prime

Question 10

Basic property of Jacobi

Answer 10

multiplicative for both entries