Factor a given polynomial over two rings

Show that p(x) = x^4 + 1 is irreducible over \mathbb{Q} and reducible over \mathbb{Q}[i].


Note that p(x+1) = x^4 + 4x^3 + 6x^2 + 4x + 2 is Eisenstein at 2 and thus irreducible; so p(x) is irreducible over \mathbb{Q}. On the other hand, p(x) = (x^2+i)(x^2-i). Thus p(x) is reducible over \mathbb{Q}[i].

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