## 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 $\mathbb{Q}[i]$.