We are aiming to give an overview of a model theorist's take on certain number theoretic topics of diophantine nature. Our approach is based on the study of 'small' subgroups of the multiplicative group of a field of characteristic zero. Here 'small' is quite a technical term which happens to be the model theoretic counter-part of 'finite rank'. Model theory needed for this talk is at the minimum and will be summarized in the first few minutes.
