Abstract
Carlitz characterized the number fields K with class number \(\le \)2 by the equality of the lengths of all the factorizations of every integer of K into irreducible elements. Analogously, we study the links between the order of the Pólya group \(\mathscr {P}o(K)\) of a number field K and the factorizations into irreducible elements of some rational numbers. Our main results concern quadratic fields where we prove some equivalences between, on the one hand, \(\vert \mathscr {P}o(K)\vert =1\) and uniqueness of factorizations, on the other hand, \(\vert \mathscr {P}o(K)\vert =2\) and uniqueness of lengths of factorizations. We also show how analogous results may be formulated in the case of function fields.
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Acknowledgments
The authors want to thank the anonymous referee who suggested to study the problem in the framework of the theory of factorization in monoids and proposed almost everything that is contained in Sect. 3.3.
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Adam, D., Chabert, JL. (2016). About Number Fields with Pólya Group of Order \(\le \)2. In: Chapman, S., Fontana, M., Geroldinger, A., Olberding, B. (eds) Multiplicative Ideal Theory and Factorization Theory. Springer Proceedings in Mathematics & Statistics, vol 170. Springer, Cham. https://doi.org/10.1007/978-3-319-38855-7_2
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DOI: https://doi.org/10.1007/978-3-319-38855-7_2
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