As seen in Chapter 10, there are infinitely many algebraic number fields in which the uniqueness of factorization of integers fails (it is not known whether there are infinitely many fields with uniqueness of factorization). As already mentioned, of the two approaches to remedy the situation, the introduction of ideals has proved extremely successful while the continued extension of fields led to a certain disappointment (infinite class field towers). For this reason, we here interrupt the study of algebraic number fields, to develop the theory of ideals in rings of algebraic integers.
KeywordsPrime Ideal Ideal Theory Algebraic Number Class Number Principal Ideal
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