Given a field extension $\mathbb{F}_{p^n}$ and some element $\alpha \in \mathbb{F}_{p^n}$ which is not contained within any proper subfield of $\mathbb{F}_{p^n}$, is there a lower bound on the order of $\alpha$?
I understand that the nonzero elements of a finite field form a cyclic group generated by some primitive element $\beta \in \mathbb{F}_{p^n}$. However, if we don't know whether $\alpha$ is primitive, what can we say about its order (without actually computing anything)?
Answer
From the fact that $\alpha$ is not contained within any proper subfield, we know it is not fixed by any power of Frobenius, so that the elements $\alpha, \alpha^p, \alpha^{p^2},...,\alpha^{p^{n-1}}$ are all distinct, i.e. there exist $n$ distinct powers of $\alpha$ not equal to the identity, so $\alpha$ must have order greater than $n$.
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