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Exercises - Legendre's Symbol

1. Prove the following, assuming that $p$ is an odd prime, while $a$ and $b$ are integers not divisible by $p$:

   • If $a \equiv b \pmod{p}$, then $\displaystyle{\left( \frac{a}{p} \right) = \left( \frac{b}{p} \right)}$

   • $\displaystyle{\left( \frac{a}{p} \right) \left( \frac{b}{p} \right) = \left( \frac{ab}{p} \right)}$

   • $\displaystyle{\left( \frac{a^2}{p} \right) = 1}$

2. Evaluate $\displaystyle{\left(\frac{111}{991}\right)}$.