If a is a quadratic residue, and ab is a quadratic residue, how can I show that b is also a quadratic residue? Would appreciate a hint.
So far I thought about the problem a little and I have:
$a^2$ = x mod n and $b^2$ = xy mod n, so $(\frac{b}{a})^2$ = y mod n, so I just have to show that $\frac{b}{a}$ is an integer. Is that a valid approach so far?
Try proving that the product of a QR with a QNR must be a QNR.
Let $a$ be a QR and $b$ be a QNR mod $p$. Then $ab=x^2b\pmod{p}$. Assume that this is a QR and conclude that $b$ must have been a QR for a contradiction.
Now that the OP has found the answer, I'll include on for the sake of my own neurosis.
Let $ab=x^2b\pmod{p}$ as described above. Since the integers mod $p$ form a field, every non-zero element is investable, by hypothesis $a,b$ are not zero (since QR and QNR are defined as terms on the unit group), so $\exists y$ such that $xy=1$. Then $y^2(ab)=b$. If $ab=z^2$, then we would have $b=(yz)^2$, a contradiction since $b$ is a QNR. Thus $ab$ is a QNR.