## Irreducible Polynomial [closed]

Let p be a positive prime. Show that P(x)=x^p - x + 1 is irreducible over Q[x].

Note: The problem is not supposed to be solved using Galois' Theory.

Note 2: I think there's an "olympic solution" (using Gauss' Lemma). So let's try to find it (I actually don't know it).

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 Not suitable for this site, see the FAQ. You might get some responses if you try posting your question on math.stackexchange.com. – J.C. Ottem Jan 21 2012 at 21:15 The polynomial is irreducible mod p? It looks like you're asking for a technically elementary proof of a weaker result than what follows very quickly from the theory of finite fields. (Which might be called Galois theory by some, but scarcely needs the general theory.) – Charles Matthews Jan 21 2012 at 21:24