Show that $x^4+1$ is reducible in p-adic numbers $\mathbb{Q}_p$ for p>2 prime.
This is a homework problem for algebraic number theory but I'm having trouble getting started. Do I use induction in general, or show this holds for $p \equiv 1,3$ (mod 4)?
Any help would be appreciated!
Solution 1:
(1) Use the fact that $$X^4+1 = (X^2+\sqrt{-1})(X^2-\sqrt{-1}) = (X^2+\sqrt{2}X+1)(X^2-\sqrt{2}X+1) = (X^2+\sqrt{-2}X-1)(X^2-\sqrt{-2}X-1),$$ to show that $X^4+1$ is reducible in $\mathbb{F}_p[X]$ (even for $p=2$). You may need the law of quadratic reciprocity.
(2) Conclude with Hensel Lemma.