Editing Dedekind zeta function (section)

==Arithmetically equivalent fields==

Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. {{harvs|txt | last1=Bosma | first1=Wieb | last2=de Smit | first2=Bart | year=2002 | volume=2369 }} used [[Gassmann triple]]s 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.

{{harvtxt|Perlis|1977}} showed that two [[number field]]s ''K'' and ''L'' are arithmetically equivalent if and only if all but finitely many prime numbers ''p'' have the same [[splitting_of_prime_ideals_in_Galois_extensions|inertia degree]]s in the two fields, i.e., if <math>\mathfrak p_i</math> are the prime ideals in ''K'' lying over ''p'', then  the tuples <math>(\dim_{\mathbf Z/p} \mathcal O_K / \mathfrak p_i)</math> need to be the same for ''K'' and for ''L'' for almost all ''p''.