L.S.,
I would like to factor $(3)$ in $K = \mathbb{Q}(\alpha)$ where $\alpha$ is a root of $f = X^4 + 4X^2 + 2$. I need this factoring as a part of an exercise I need to do from my course on algebraic number theory. I believe that I am on the right track but something perculiar is going on. Any help would be greatly appreciated!
I already showed that $\mathcal{O}_K = \mathbb{Z}[\alpha]$. so $3$ does not divide $[\mathcal{O}_K : \mathbb{Z}[\alpha]]$. $\alpha$ is a primitive element of $K$ with minimal polynomial $f$, and thus we may use the Kummer-Dedekind theorem to factor $(3)$ with these $\alpha, f$. We have that $X^4 + 4X + 2 \equiv X^4 + X + 2$ modulo 3, which is irreducible modulo 3. But then Kummer-Dedekind says that $(3) = (3,\alpha^4 + \alpha + 2)$. But this is weird, since according to sagemath this ideal is the whole ring of integers. It also says that $(3)$ should remain prime. What am I doing wrong? Why can't I invoke Kummer-Dedekind?
Many thanks!