= Solution
For an odd <prime number> $p$, a nonzero class $a\pmod p$ is a <quadratic residue> if $a\equiv x^2\pmod p$ for some <integer> $x$ not divisible by $p$. The <Legendre symbol> is
$$
\left(\frac ap\right)=
\begin{cases}
0,&p\mid a,\\
1,&a\text{ is a nonzero square modulo }p,\\
-1,&a\text{ is a nonsquare modulo }p.
\end{cases}
$$
When discussing solvability of $x^2\equiv a\pmod p$ without a coprimality assumption, zero is also a square. This convention matters in the final local characterization of <integer> squares.
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