Number of quadratic residues mod $p$

Let $p$ be an odd prime. If $a$ not congruent to $0$ modulo $p$, the congruence $x^2\equiv a\pmod {p}$ has two solutions or none.

This means that viewed as a function on the non-zero elements of $\mathbb{Z}_p$, the squaring function is $2$ to $1$, meaning that its range has cardinality $(p-1)/2$.

Remark: The fact that if there is a solution, there are exactly $2$, is true in all fields $F$ of characteristic different from $2$. For suppose that in the field $F$, we have $a=b^2$, where $a\ne 0$. Then the equation $x^2=b^2$ can be rewritten as $(x-b)(x+b)=0$, so its only solutions are $b$ and $-b$, which are different unless $F$ has characteristic $2$.

Let $p>2$. The map $f:\Bbb Z_p^\times\rightarrow\Bbb Z_p^\times$ given by $f(x)=x^2$ is an homomorphism, because $f(xy)=(xy)^2=x^2y^2=f(x)f(y)$. By the isomorphism theorem we know that $$ {\rm Im}(f)\simeq \Bbb Z_p^\times/\ker(f). $$ Since $\ker(f)$ is just the set of roots of $x^2-1$, it is immediate that $\ker(f)=\{\pm1\}$ and $|{\rm Im}(f)|=(p-1)/2$, thus showing that there are exactly $(p-1)/2$ non zero classes that are squares.

Also, using the fact that $\Bbb Z_p^\times$ is cyclic it is easy to show that $x\in\Bbb Z_p^\times$ is a square if and only if $$ x^{\frac{p-1}2}=1. $$ This links the above to the fact that $$ \left(\frac xp\right)=(-1)^{\frac{p-1}2}. $$

Let $p>2$ be a prime and consider the following map: $$\psi : (\mathbb{F}_p)^\times \to (\mathbb{F}_p)^\times$$ that sends $\psi(x)\equiv x^2 \bmod p$. This is a homomorphism of groups. Indeed: $$\psi(x\cdot y) \equiv (x\cdot y)^2 \equiv x^2\cdot y^2 \equiv \psi(x)\cdot \psi(y)\bmod p.$$ Moreover, the kernel has order $2$, because $\psi(x)\equiv 1 \bmod p$ if and only if $x^2\equiv 1 \bmod p$ if and only if $x\equiv \pm 1\bmod p$. (Note: a prime $p$ divides $x^2-1=(x-1)(x+1)$ if and only if $p$ divides $x+1$ or $x-1$.)

Hence, the image of $\psi$ is isomorphic to $\mathbb{F}_p^\times/\{\pm 1\}$, which has order $\frac{p-1}{2}$. Since the image of $\psi$ is exactly the subgroup of quadratic residues modulo $p$, we obtain the desired result.