Playing around with some examples of quadratic residues, I've come up with the following:
Let $q = p^n$ be an odd prime power for some odd prime $p$ and some $n \in \mathbb{N}$. Suppose $\mathbb{F}_q^* = \langle \varepsilon \rangle$ for some generator $\varepsilon$. Consider $-1$ as an element of $\mathbb{F}_q^*$. Then the following hold:
- If $q \equiv 1$ mod 4, then $-1 = a^2 \ $ for some $a \in \mathbb{F}_q^*$
- If $q \equiv 3$ mod 4, then $-1 = b^2\varepsilon \ $ for some $b \in \mathbb{F}_q^*$
Anyone have any idea if it even holds true, or why? Any literature knocking about on something like this?
It seems the first case is well known if $n=1$, that is if $q$ is itself prime, but I don't know about generalizing - like I say it's just a pattern that I've noticed. Likewise the second case is just a pattern I've noticed (from some low prime power examples) and I'm wondering if it generalizes too? Thanks!