Exercise 6. Let pinZ^(+)be an odd prime, and suppose that g is a primitive root
modulo p.
a) Prove that for any ainZ coprime to p, one has that a is a quadratic residue
modulo p if and only if a-=g^(2k)(modp) for some kinZ.
b) Use part a) to give a complete list of quadratic residues modulo 19; write them|
using positive integers between 1 and 18, listed in increasing order.
(Hint: 2 is a primitive root modulo 19.)
c) How many quadratic residues are there modulo an odd prime pinZ^(+)? What
about quadratic nonresidues?
Exercise 6. Let p e Z+ be an odd prime, and suppose that g is a primitive root modulo p. a) Prove that for any a e Z coprime to p. one has that a is a quadratic residue modulo p if and only if a = g2k (mod p) for some k E Z. b) Use part a) to give a complete list of quadratic residues modulo 19; write them| using positive integers between 1 and 18, listed in increasing order. Hint: 2 is a primitive root modulo 19.) c) How many quadratic residues are there modulo an odd prime p e Z+? What about quadratic nonresidues?