I am stuck with problem 22, chapter 3 in Marcus’ book Number Fields which says:
Suppose $\alpha^5=2\alpha+2$. Prove that the ring of integers of $\mathbb Q[\alpha]$ is $\mathbb Z[\alpha]$. Prove the same thing also if $\alpha^5+2\alpha^4=2$.
Try: Discriminant for both of them is not square free.
Let’s look at the first part. Let $R$ be the ring of integers of $\mathbb Q[\alpha]$. Let $\mathfrak p$ be a maximal ideal of $R$ containing $2$. Then $\alpha^5=2(1+\alpha)\in \mathfrak p$, hence $\alpha\in \mathfrak p$. Moreover $1+\alpha\notin \mathfrak p$, so $2\in \mathfrak p^5R_\mathfrak p$. This implies that the ramification index at $\mathfrak p$ is at least $5$. But $\mathbb Q[\alpha]$ has degree $5$, so the ramification index is exactly $5$, the residue extension is trivial and there is only one prime above $2$. By Problem 21 just before the one you are considering, $2^4$ divides the discriminant of $R$.
Now the discriminant of $X^5-2X-2$ is $2^18.104.22.168$. So it is equal to the discriminant of $R$ and therefore $\mathbb Z[\alpha]=R$.
The seconde part should be solved similarly.
Edit To see the ramification index at $\mathfrak p$ is at least $5$
without using localization (I don’t know how things are organized in Marcus), we can just compare ideal decompositions of $\alpha^5 R$ and $2(1+\alpha)R$, noticing that no prime ideal can contains at the same time $\alpha$ and $1+\alpha$.