Fitting Ideals in Number Theory and Arithmetic

Abstract We describe classical and recent results concerning the structure of class groups number fields as modules over Galois group. When presenting more modern developments, we can only hint at much broader context very powerful general techniques that are involved, but endeavour to give complete statements or least examples where feasible. The timeline goes from a result proved in 1890 (Stickelberger’s Theorem) (2020) breakthrough: proof Brumer-Stark conjecture by Dasgupta Kakde.