Questions about the square root of $a$ in $\mathbb{F}_p$.

Question

How to prove that there is a square root of $3$ in $\mathbb{F}_p$ if and only if $p \equiv 1 $ or $11 \pmod {12}$? Thank you very much.

Answer 1

Highlights for $\;p>3\;$:

$$\sqrt3\in\Bbb F_p\iff \exists\,x\in\Bbb F_p\;\;st.\;\;x^2=3\iff \left(\frac3p\right)=1$$

Using now quadratic reciprocity , we get that

$$\left(\frac3p\right)=\begin{cases}\;\;\left(\frac p3\right)&,\;\;p=1\pmod 4\\{}\\\!-\left(\frac p3\right)&,\;\;p=3\pmod 4\end{cases}$$

and now checking each case above, and using that $\;\left(\frac p3\right)=1\iff p=1\pmod 3$ we get what we want. For example, check that

$$p=3\pmod 4\implies \left(\frac 3p\right)=1\iff p=2=-1\pmod 3\iff$$

$$\iff p=11\pmod{12}\ldots$$