Let \(K = \mathbb{F}_{p^n}\) be the finite field extension of degree \(n\) over \(F = \mathbb{F}_p\). (1) What is the Frobenius endomorphism \(\operatorname{Frob}_p: K \to K\), and how does it generate the Galois group \(\operatorname{Gal}(K/F)\)? (2) Viewed as an \(\mathbb{F}_p\)-linear transformation \(T: K \to K\) on the \(n\)-dimensional vector space \(K \cong \mathbb{F}_p^n\), what are its minimal and characteristic polynomials?
Solution
The Frobenius map is \[ F(x)=x^p. \] It is an \(\mathbb F_p\)-linear field automorphism of \(K=\mathbb F_{p^n}\), and \[ F^k(x)=x^{p^k}. \] Its fixed field is \(\mathbb F_p\), and \(F\) has order \(n\); hence \[ \operatorname{Gal}(\mathbb F_{p^n}/\mathbb F_p)=\langle F\rangle\cong C_n. \]
Viewed as an \(\mathbb F_p\)-linear operator on the \(n\)-dimensional space \(K\), the normal basis theorem gives \(\alpha\in K\) such that \[ \alpha,F\alpha,\dots,F^{n-1}\alpha \] is a basis. Relative to this basis, \(F\) is the cyclic shift, so \(\alpha\) is a cyclic vector and the minimal polynomial has degree \(n\). Since \(F^n=I\), its minimal polynomial divides \(x^n-1\); therefore \[ \mu_F(x)=x^n-1. \] The characteristic polynomial has degree \(n\) and is divisible by the minimal polynomial, so \[ \chi_F(x)=x^n-1. \]