I am trying to solve some exercises from last year's exam, and there is this exercise I am stuck. Can anyone help me please?
Let $\alpha$ be a root of $f(X) = X^4+4X+2$, and $K = Q(\alpha)$. show that $[O_K:\mathbb Z[\alpha]]\le 2^4$. Show that the inclusion $\mathbb Z[\alpha]\rightarrow O_K$ induces isomorphisms from $\mathbb Z[\alpha]/\alpha \mathbb Z[\alpha] \rightarrow O_K/\alpha O_K$ and $\mathbb Z[\alpha]/2 \mathbb Z[\alpha] \rightarrow O_K/2O_K$.
The first part is clear, following from $disc(f) = 2^8. 19$ and $disc(f) = disc(O_K).[O_K:\mathbb Z[\alpha]]^2$. It's also clear that the induced morphisms are injective. I am having problem with showing why they are surjective. Also, how do we find how the prime 2 factors in $O_K$? 2 might divide index, and we will not be able to use Dedekind's theorem.