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Exercise from Hungerford's Algebra, Page 166, Exercise 8:

(a) Let $c$ be an element of a field $F$ of characteristic $p$ ($p$ prime). Then $x^p-x-c$ is irreducible in $F[x]$ if and only if $x^p-x-c$ has no root in $F$.

(b) If $\operatorname{char}F = 0$, then part (a) is false.

I don't understand why it is false when $\operatorname{char}F = 0$. What is an example of such a field?

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  • I don't understand what are you asking: the claim there is true . For any polynomial $;p(x)\in F[x],,,F;$ any field, of degree less than four, it is reducible over $;F;$ iff it has a root in $;F;$ . – DonAntonio Mar 16 '16 at 21:24
  • But if the polynomial is of degree larger than 4? –  Mar 16 '16 at 21:29
  • The book says the proposition does not necessarily hold in field of characteristic 0. I need a counter example. –  Mar 16 '16 at 21:30
  • Right, I missed that the degree isn't necessarily less than four. – DonAntonio Mar 16 '16 at 21:31
  • The proposition is true for characteristic p( by Artin-Schreier), but I don't know why it is false for characteristic 0. –  Mar 17 '16 at 01:50
  • I can't think of any counterexample of the precisely exact form you want. – DonAntonio Mar 17 '16 at 10:37
  • Here one can find this example: $x^5-x+15=(x^2+x+3) (x^3-x^2-2 x+5) \in \mathbb Q[x]$. – user26857 Apr 13 '16 at 23:30

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