My work:
How do continue with part $b$, I'm so confused.
2026-03-25 19:10:57.1774465857
Legendre Symbol by Fermat's little theorem.
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All elements of $\mathbf Z_p^\times$ satisfy the equation $x^{p-1}=(x^q)^2\equiv=1\mod p,\;$ i.e. $\;x^q\equiv 1$ or $x^q\equiv-1$. Now, all squares satisfy $x^q\equiv1$ by Lil' Fermat, and there are $q$ of them.As there can't be more than $q$ solutions to this equation, any element $c$ satisfying $c^q\equiv 1$ is a square.