Class-Group and Class-Number
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We now use the geometric ideas that we have developed to build further insight into the property of unique factorization. We already know that in the ring of integers of any number field, factorization into primes is unique if and only if every ideal is principal. We now refine this statement to give a quantitative measure of how non-unique factorization can be. To do this we use the fractional ideals introduced in Chapter 5. The class-group of a number field is defined to be the quotient of the group of fractional ideals by the (normal) subgroup of principal fractional ideals; the class-number is the order of this group. This gives the required measure: factorization in a ring of integers is unique if and only if the corresponding class-number is 1. If the class-number is greater than 1, factorization is non-unique. Intuitively, the larger the class-number is, the more complicated the possibilities for non-uniqueness are. In a sense, prime factorization becomes ‘less unique’ as the class-number increases.