The Ideal Class Group
نویسنده
چکیده
When we form a finite algebraic extension of Q, we are not guaranteed that the ring of integers, O, in our extension will be a unique factorization domain (UFD). We can obtain a measure of how far O is from being a UFD by computing the class number which is defined as the order of the ideal class group. This paper describes the ideal class group and provides examples of how to compute this group. First, necessary terms are defined, such as norms of elements in a field extension and the ring of integers of a field. Then, we introduce the concept of fractional ideals and exploit the properties of Dedekind domains to show that the non-zero fractional ideals of a Dedekind domain form a group. This leads to a definition of the ideal class group and a proof of its finiteness. We then make these concepts more concrete by computing ideal class groups of quadratic field extensions. We finish by discussing the problem of computing discriminants given a class number.
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