TITLE: Show that $x^4 + 8$ is irreducible over Z
QUESTION [16 upvotes]: Is there an easy way to show that $x^4+8$ is irreducible over $\mathbb Z$ without trying to write it as a product of  polynomials of lower degrees?

REPLY [2 votes]: The roots of $x^4+8$ have absolute values $2^{3/4}$. The constant term of a factor of degree $d$ would have absolute value $2^{3d/4}$. But this can't be integral for $d=1,2,3$.