Answered: Let p be a prime and let a ∈ Fp \{0}.…

Let p be a prime and let a ∈ Fp \{0}. Prove that the Galois group of f(x) = xp − x +a over Fp is cyclic as follows: Let K be an algebraic closure of Fp and let α ∈ K be a root of f(x)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Let p be a prime and let a ∈ Fp \{0}. Prove that the Galois group of f(x) = xp − x +a over Fp is cyclic as follows: Let K be an algebraic closure of Fp and let α ∈ K be a root of f(x) and let σ: Fp(α) −→ Fp(α) be the isomorphism given by σ(α) = α+1. Show that o(σ) = p.

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