Let $a$ be a nonzero element of $\mathbb{F}_q$. Then $a = \omega^i$ for some primitive element $\omega$ in $\mathbb{F}_q$. This means that the equation $x^n=a$ has a solution in $\mathbb{F}_q$ if and only if $gcd(n, q-1)|i.$ I am asked to prove this fact. I have no idea how to start at all. I can start off with the forward implication by setting $x^n = \omega^i$ but I do not know how to proceed at all. I can use the logic that since $\omega$ is a generator, the order of $\omega = q- 1$ but I am not sure what to do. Can anyone please help me get started off? I missed my class.
2026-04-09 03:33:24.1775705604
Proof about finite fields and primitive elements
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