Evaluate parts (c) and (d) of Example 7.19.
We want to determine
$$ (-1)^{(\frac{p-1}{2}) \times (\frac{q-1}{2})} $$
(c) $p \equiv 1 \pmod 4$ and $q \equiv 3 \pmod 4$
(d) $p \equiv 3 \pmod 4$ and $q \equiv 1 \pmod 4$
(c) We write $p=4x + 1$ and $q=4y+3$ for some integers $x,y$. And so
$$ (-1)^{(\frac{p-1}{2}) \times (\frac{q-1}{2})} = (-1)^{(2x) \times (2y+1)} = 1$$
because the index is even.
(d) We write $p=4x + 3$ and $q=4y+1$ for some integers $x,y$. And so
$$ (-1)^{(\frac{p-1}{2}) \times (\frac{q-1}{2})} = (-1)^{(2x+1) \times (2y)} = 1$$
because the index is even.