Prove that if $ab \equiv 1 \pmod{p}$ and $a$ is quadratic residue mod $p$, then so is $b$
where $p$ is odd prime, and $(a,p) = (b,p) = 1$.
Besides $b$ is the inverse of $a$, what else does this $ab \equiv 1 \pmod{p}$ tell us? A hint would be greatly appreciated.
Thank you,
Best Answer
The fact that $b$ is the modular inverse of $a$ is more than enough.
If these were the rationals, if $r^2 = a$ (with $r$ and $a$ rationals), then is $\frac{1}{a}$ a square? What number works as the square of $\frac{1}{a}$?
Now do the same thing, but modulo $p$.