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Unit groups of quotients of number fields. (English) Zbl 1475.11195

Let \( K\) be a number field , \(O_{K}\) it’s ring of integers and let \(I \subset O_{K}\) be a non zero ideal. In this paper, the authors are interested in giving an explicit description, of the unit group \((O_{K}/I)^{*}\). In the cases of the quadratic and certain cubic fields, A. Harnchoowong and P. Ponrod in the paper [Commun. Korean Math. Soc. 32, No. 4, 789–803 (2017; Zbl 1409.11092)] calculated such explicit description for monogenic cubic fields of square free discriminant. In this paper, the authors do not assume any hypothesis on the degree, discriminant or the structure of the ring of integers: and they give a cohesive result that includes all the cases, except when \(p = 3\) ramifies, described in their last paper.

MSC:

11R21 Other number fields
11R27 Units and factorization

Citations:

Zbl 1409.11092
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Full Text: DOI

References:

[1] A. Harnchoowong and P. Ponrod, Unit groups of quotient rings of integers in some cubic fields, Commun. Korean Math. Soc. 32 (2017), no. 4, 789-803. https://doi.org/10.4134/CKMS.c160254 · Zbl 1409.11092 · doi:10.4134/CKMS.c160254
[2] G. Mantilla-Soler, Integral trace forms associated to cubic extensions, Algebra Number Theory 4 (2010), no. 6, 681-699. https://doi.org/10.2140/ant.2010.4.681 · Zbl 1201.11100 · doi:10.2140/ant.2010.4.681
[3] J. Neukirch, Algebraic Number Theory, translated from the 1992 German original and with a note by Norbert Schappacher, Grundlehren der Mathematischen Wissenschaften, 322, Springer-Verlag, Berlin, 1999. https://doi.org/10.1007/978-3-662-03983-0 · Zbl 0956.11021
[4] A. · JFM 41.0245.01 · doi:10.1090/S0002-9947-1910-1500859-8
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