How do I prove that x^p-x+ax^p-x+a is irreducible in a field with pp elements when a\neq 0a\neq 0?

Let p be a prime. How do I prove that x^p-x+a is irreducible in a field with p elements when a\neq 0?

Right now I’m able to prove that it has no roots and that it is separable, but I have not a clue as to how to prove it is irreducible. Any ideas?

Answer

x \to x^p is an automorphism sending r to r-a for any root r of the polynomial. This operation is cyclic of order p, so that one can get from any root to any other by applying the automorphism several times. The Galois group thus acts transitively on the roots, which is equivalent to irreducibility.

Attribution
Source : Link , Question Author : MathTeacher , Answer Author : zyx

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