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Abstract: Outlines my work at the 1999 REU program at LSU. I searched for and found two number fields with the same zeta function but with class numbers differing by a factor of five.   Examples of the class numbers of two number fields with the same zeta function differing by a small factor have been found, so if there is some maximum ratio between two such class numbers, it would certainly be quite interesting. It's generally expected that these arithmetically equivalent number fields can have class numbers differing by any integer factor, but this hasn't been proven.   My adviser, Dr. Perlis, presented me with a way to produce pairs of arithmetically equivalent number fields which had some hope of differing in class number by an integer factor. I went to town calculating their class numbers using the program KANT, which had been made that year. As far as we know, I found the first example of a class number ratio of 5; examples had previously been found for 2 and 3, but the computational requirements increase very quickly as one tries to find higher and higher factors. The whole experience was really fun. |