How to check if $2$ is a square $\mod 3$?

Question

I don't think I can use the Legendre or Jacobi symbol here because $2$ is an even prime. I'm not sure I've learned methods to deal with $2$ even though I know how to use quadratic reciprocity, it only works with odd numbers I think.

Answer 1

In a more general context $2$ is a square mod $p$ where $p$ is an odd prime if and only if :

$$\begin{pmatrix}2\\p\end{pmatrix}=(-1)^{\frac{p^2-1}{8}}\text{ is } 1$$

In your case, because $3^2-1=9-1=8$ the answer is no.

Answer 2

$0^2 = 0 \equiv 0 \pmod3$

$1^2 = 1 \equiv 1 \pmod3$

$2^2 = 4 \equiv 1 \pmod3$

So, no.