Suppose $\theta$ is a root of a irreducible monic polynomial $f$ of degree $n$. (In practice, I would like to deal with $n=3$ case.) Then, define the ideal class group of $\mathbb{Z}[\theta]$, $C(\mathbb{Z}[\theta])$ by the set of invertible fractional ideals modulo principal ideals.
Given a polynomial $f$ and a fractional ideal $I$ of $\mathbb{Z}[\theta]$, is there any way to decide I is principal or not?