Two number fields K and L are said to be arithmetically equivalent if, for almost every prime number p, the factorizations of p in the rings of integers of K and L are analogous (in a precise sense that will be explained). A completely similar definition can be given for finite extensions of a function field F(T), where F is a finite field.
In this talk we discuss the concept of arithmetic equivalence in both contexts, focusing on the similarities and the differences between the two cases. In particular, we will show a group-theoretic analogue of the problem and we will explain the relation between arithmetic equivalence and equality of certain zeta function (the classical Dedekind zeta function for number fields, a more complicated function for function fields). Finally, we will show how to produce examples of equivalent but not isomorphic fields in both contexts.
