Irreducible over $\mathbb Q[x]$ but reducible over $\mathbb F_p$ for all $p$

Question

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 $$f(x)=x^4+1\in K[x]\space\text {is irreducible over }K\iff K=\mathbb Q.$$