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Considering a prime number $p$ and a natural number $n \geqslant 1$, I want to prove that there exists an automorphism $f \in GL(E)$ with $E = (\mathbb{F}_p)^n$ such that $f$ only stabilizes $E$ and $\{0\}$. Moreover, we know that this is equivalent to finding an irreducible polynomial $P(X) \in \mathbb{F}_p[X]$ of degree exactly $n$. However I don't know whether it is always possible.

thomas martinelli
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    It is always possible to find such polynomials. Try looking up a proof of the existence of finite fields—the existence of an irreducible polynomial in $\mathbb{F}_p[x]$ of degree $n$ is equivalent to the existence of a finite field with $p^n$ elements. – Kenanski Bowspleefi Feb 06 '25 at 09:22

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