Square root of $-1$ over a finite field

Question

It is known that the equation $x^2 \equiv -1 \pmod{p}$, where $p$ is an odd prime number, has a solution iff $p = 4k +1$ for some natural $k$.

Does it exist a similar characterization for a general finite field $F_q$, where $q = p^n$?

Thanks a lot for your help!

Answer 1

Suppose that $x^2+1$ has a root in $\mathbb{F}_q$, where $q=p^n$. Recall that $\mathbb{F}_q$ is a field extension of $\mathbb{F}_p$ of degree $n$, and that

$\mathbb{F}_{p^a}$ is a subfield of $\mathbb{F}_{p^b}$ if and only if $a$ divides $b$.

If $p=2$ or $p=4k+1$, a root exists in $\mathbb{F}_p \subset \mathbb{F}_q$.

Otherwise, such a root does not exist in $\mathbb{F}_p$. Then the splitting field of $x^2+1$ over $\mathbb{F}_p$ is $\mathbb{F}_{p^2}$. So the root is in $\mathbb{F}_q$ if and only if $\mathbb{F}_{p^2} \subset \mathbb{F}_q$ (is a subfield of) if and only if $n$ is even.