Arithmetically equivalent fields Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. used Gassmann triples to give some examples of pairs of non-isomorphic fields that are arithmetically equivalent. In particular some of these pairs have different class numbers, so the Dedekind zeta function of a number field does not determine its class number. showed that two number fields K and L are arithmetically equivalent if and only if all but finitely many prime numbers p have the same inertia degrees in the two fields, i.e., if are the prime ideals in K lying over p, then the tuples need to be the same for K and for L for almost all p. Notes References Section 10.5.1 of Zeta and L-functions Algebraic number theory