Despite appearances, this is not a question on computational aspects of number theory. The background is as follows. I once asked a number theorist about what he considered to be the most important unsolved problems in arithmetic geometry. He told me about a few, but along with some well-known problems he told me the following one also:
How to determine conceptually when a number is squarefree or not?
When I protested that this sounded like a computational question, he told me that no, this is not so, and demonstrated that this has a rather nice solution for the ring of polynomials over a field, which is in many senses analogous to the ring of integers. Take the polynomial, take its derivative and compute the gcd using the Euclidean algorithm. But for the ring of integers there is nothing analogous to the derivative, and he wanted a solution of the problem by constructing a good notion of a differential in this case.
Question: What are the known investigations along this line? What well-known topics in arithmetic geometry are related to this? And what would be some other interesting consequences of a successful development of such a method?
Any other comments that might enlighten me further would be received with gratitude.