To solve this question I have almost finished the proof but I need a little detail to be rigorous.
Let $K$ be the prime fields $\mathbb Q$ or $\mathbb F_p$. Prove that $$f(x)=x^4+1\in K[x]\space\text {is irreducible over }K\iff K=\mathbb Q.$$
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2And which detail would that be? – Hagen von Eitzen Apr 17 '16 at 14:46
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8See http://math.stackexchange.com/questions/77155/reducible-polynomial-modulo-every-prime?rq=1 – Captain Lama Apr 17 '16 at 14:46
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@HagenvonEitzen: I want to apply the following proposition (not very known I guess): $f$ of degree $n$ is irreducible over $\mathbb F_p$ iff $f$ has no root in the extension of $\mathbb F_p$ of degree $\le \frac n2$. I must apply the 8-roots of unity for finish but a small final detail makes me doubt a little. – Piquito Apr 17 '16 at 15:18
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@ProbablyWrong: I am "probably wrong" dear friend in a little detail as I said. See the comment to Hagen von Eitzen, please. (It is forbidden in StackExchange a comment to two persons at the same time). – Piquito Apr 17 '16 at 15:26