Congruent Integers are of same Quadratic Character

Theorem

Let $p$ be an odd prime.

Let $a \in \Z$ be an integer such that $a \not \equiv 0 \pmod p$.

Let $a \equiv b \pmod p$.

Then $a$ and $b$ have the same quadratic character.

Proof

Let $a \equiv b \pmod p$.

Then by Congruence of Powers:

$a^2 \equiv b^2 \pmod p$

Hence:

$x^2 \equiv a \pmod p$ has a solution if and only if $x^2 \equiv b \pmod p$.

Hence the result.

$\blacksquare$