Arithmetically Equivalent Fields
Two fields are called arithmetically equivalent if they have the same Dedekind zeta function. Wieb Bosma and Bart de Smit (2002) 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.
Read more about this topic: Dedekind Zeta Function
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