Two number fields are called arithmetically equivalent if their Dedekind zeta functions coincide. Thanks to the work of R. Perlis, we know that much of the arithmetic information of a number field is encoded in its zeta function. By interpreting the Dedekind zeta function as the Artin L-function attached to a certain Galois representation, we see how all the information mentioned above can be recovered in a very natural way. Moreover, we will show how this approach leads to new results. Going further, we will see how from zeta functions we can connect with trace forms and we will explore the classification power of integral trace forms.
Guillermo Mantilla Soler