Recall that a number field $K$ is said to be monogenic if its ring of integers is of the form $\mathbb{Z}[\alpha]$, or equivalently, if it has a power integral basis. There are many references one can consult about the number fields case, this type of fields has been studied in particular in connection to diophantine equations. My interest is in the global function fields analogue, say finite extensions $F/\mathbb{F}_q[T]$, where $\mathbb{F}_q$ is the finite field with $q$ elements. The "ring of integers" in this case is the ring of elements regular away from a fixed place $\infty$. I have been googling around and did not find as many sources (books, papers, ...) on that as I expected. My naive question is: is it not as interesting as the number fields case? If so, what might be the reason?
In the number fields case, Dedekind gave a criterion for monogeneity (or non-monogeneity) of number fields. I am wondering if there is anything analogous for function fields.
Thank you.