An equation over $\Bbb F_{3^k}$

Question

Does the equation $$x^2=2=-1$$ have solutions in any extension field of $\Bbb F_3$?

Answer 1

The polynomial $X^2 + 1$ is irreducible over $\mathbb{F}_3$, as it has no root. (Check by direct computation.)

The splitting field of $X^2 + 1$ is thus a degree $2$ extension of $\mathbb{F}_3$. It is thus, $\mathbb{F}_{3^2}$.

It follows that $X^2 + 1$ has a root in $\mathbb{F}_{3^k}$ if and only if $k$ is even.

Answer 2

Every field has an algebraic closure, and by definition the algebraic closure of $\mathbb F_3$ will contain solutions to any polynomial equation (of positive degree) you throw at it.

In your particular case, however, we don't need to go that far. Since $x^2+1$ is irreducible over $\mathbb F_3$, $\mathbb F_9$ can be constructed as $\mathbb F_3[X]/\langle X^2+1\rangle$, and thus the image of $X$ in that quotient will satisfy $X^2=-1$.