A General Method

There are numbers fields $K$ such that $\O_K$ is not of the form $\mathbf{Z}[a]$ for any $a\in K$. Even worse, Dedekind found a field $K$ such that $2\mid [\O_K : \mathbf{Z}[a]]$ for all $a\in\O_K$, so there is no choice of $a$ such that Theorem 4.2.3 can be used to factor $2$ for $K$ (see Example 4.3.2 below).