Let p be an odd prime Prove if a is a quadratic residue modu

Let p be an odd prime. Prove if a is a quadratic residue modul p, then its multiplicative inverse b (i.e. ab l(mod p)) is also a quadratic residue modulo p.

Solution

a is a quadratic residue modulo p

ie there is some integer x so that

x^2=a modulo p

Multipllying by b^2 gives

x^2*b^2=a*b^2 modulo p

(bx)^2=(a*b)*b modulo p

(bx)^2=1*b=b modulo p

HEnce, proved