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8] Theorem: The group G = AutFq Fqn of automorphisms of Fqn trivial on Fq is cyclic of order n, generated by the Frobenius element F(α) = αq. Proof: First, we The Galois group is cyclic and is generated by the Frobenius automorphism σ(x) = xp, x ∈ E. Proof. E is a splitting field for a separable polynomial over Fp, means that we can make sense of the action of the Frobenius endomorphism σp Theorem 1.1 (Fundamental theorem of Galois theory for finite extensions).In other words, the Frobenius elements are uniformly distributed over the conjugacy classes of the Galois group. Corollary. Let K be a number field, k a