Entropy and Følner function in algebras

We introduce the notion of entropic Følner function for algebras and we study its relation with the isoperimetric profile and the lower transcendence degree.Under the assumption of a technical conjecture we use the Shannon inequality to derive a theorem on the lower transcendence degree of domains and division algebras. We finally discuss its relation with some old conjectures of M. Artin, L. Small and J. Zhang.