Characteristic and minimal polynomials of the Frobenius automorphism

Let \(F=\mathbb F_{p^n}\) and let \[ \Phi:F\to F,\qquad \Phi(x)=x^p, \] viewed as an \(\mathbb F_p\)-linear map.

By the normal basis theorem for the finite Galois extension \(F/\mathbb F_p\), there exists \(\alpha\in F\) such that \[ \alpha,\Phi(\alpha),\ldots,\Phi^{n-1}(\alpha) \] is an \(\mathbb F_p\)-basis of \(F\). Since \(\Phi^n=\mathrm{id}\), the matrix of \(\Phi\) in this basis is the permutation matrix of the \(n\)-cycle \[ \alpha\mapsto\Phi(\alpha)\mapsto\cdots\mapsto\Phi^{n-1}(\alpha)\mapsto\alpha. \] Hence its characteristic polynomial is \[ \chi_\Phi(X)=X^n-1. \]

The vector \(\alpha\) is cyclic for \(\Phi\): the vectors \[ \alpha,\Phi(\alpha),\ldots,\Phi^{n-1}(\alpha) \] are linearly independent. Therefore no nonzero polynomial of degree \(<n\) annihilates \(\Phi\). Since \(\Phi^n-I=0\), the minimal polynomial has degree \(n\) and divides \(X^n-1\), so \[ m_\Phi(X)=X^n-1. \] Thus both the characteristic and minimal polynomials of Frobenius are \(X^n-1\).