product of quadratic residue and quadratic non residue.

Question

So QNR * QNR = QR

QR * QR = QR


QNR * QR = QNR

So set T is QNR of 23, but how do I go about answering this question?

Answer 1

Since $T \neq \emptyset$ there exists some $a \in T$. fix such an element.

Define $f : S \to T$ via $f(x)=ax$. Then, $f$ is well defined, and clearly 1-1. We claim that $f$ is also onto.

Indeed, let $y \in T$. Then, since $a \in T$ and $y \in T$, you get by repeated application of (ii) and (iii) that $a^{21}y \in S$. Then $y=f(a^{21}y)$.

This shows that $|S|=|T|=11$.

Finally, for each $x \in \{ 1,2,.., 22\}$ we have $x^2 \in S$.

This shows that $S$ must contain all 11 quadratic residues.... Since $|S|=11$, $S$ consists exactly of the 11 quadratic residues, and hence the non-residues must be in $T$.