TITLE: Why a particular ring of integers is not generated by a single element
QUESTION [5 upvotes]: It says here in the Sage documentation that the ring of integers in the number field obtained from
$$f(x) = x^3 + x^2 - 2x + 8$$
is not generated by a single element.  How would one go about showing that this is the case?

REPLY [6 votes]: Here is a relatively short proof:
If $O_K=\mathbb Z[\alpha]$ and $f$ is the minimal polynomial of $\alpha$, then we can find the factorization of $(2)$ to  prime ideals in $O_K$ by factoring $\bar f\in\mathbb F_2[x]$ to irreducible polynomials ($\bar f$ denotes the reduction of $f\in\mathbb Z[x]$ mod $2$); if $\bar f =\prod h_i^{e_i}$ then $(2)=\prod P_i^{e_i}$. Whatever the cubic polynomial $f$ is, we cannot get $(2)=P_1P_2P_3$, as that would mean that $\bar f$ has $3$ different roots in $\mathbb F_2$, which has only two elements.
However, in reality $(2)=P_1P_2P_3$. It is equivalent to $g(x):=x^3+x^2-2x+8$ having $3$ roots in $\mathbb Q_2$, and indeed $g(1)\equiv0$ mod $8$ and $g(2)\equiv g(4)\equiv0$ mod $16$, while $g'(1)\equiv 1$ mod $2$ and $g'(2)\equiv g'(4)\equiv 2$ mod $4$, so Hensel's lemma (i.e. Newton's method for computing roots) will give us $3$ roots (close to $1,2,4$).