Abstract
We know already that the condition h(K) = 1 is both necessary and sufficient for the uniqueness of factorization in R K . This shows that fields with trivial class-group can be characterized arithmetically in terms of factorization properties. The discovery made by Carlitz that one can similarly characterize in a simple way fields with class-number 2 gave rise to the thought that it might be possible to obtain a similar description of fields with a given class-number, or class-group. We start with Carlitz’s result. To be able to state it we need a simple definition: if a ∈ R K is neither zero nor a unit, and a = αl ... α k is a factorization of a into irreducible elements of R K ,then k is called the length of this factorization.
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© 2004 Springer-Verlag Berlin Heidelberg
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Narkiewicz, W. (2004). Factorizations. In: Elementary and Analytic Theory of Algebraic Numbers. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-07001-7_9
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DOI: https://doi.org/10.1007/978-3-662-07001-7_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-06010-6
Online ISBN: 978-3-662-07001-7
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