Kientéga, Gérard; Tapsoba, Théodore; Tougma, Charles Wend-Waoga Pólya fields that are splitting fields of \(S_3\)-fields. (English) Zbl 1426.11124 JP J. Algebra Number Theory Appl. 41, No. 2, 219-243 (2019). Summary: A number field is called a Pólya field if the module of integer-valued polynomials over its ring of integers has a regular basis. A cubic field is a \(S_3\)-field if the Galois group of its splitting field is isomorphic to the symmetric group \(S_3\). In this paper, we characterize Pólya fields that are splitting fields of \(S_3\)-fields. MSC: 11R16 Cubic and quartic extensions Keywords:cubic fields; regular basis; principal ideals; ramification PDFBibTeX XMLCite \textit{G. Kientéga} et al., JP J. Algebra Number Theory Appl. 41, No. 2, 219--243 (2019; Zbl 1426.11124) Full Text: DOI References: [1] S. Alaca, p-integral bases of cubic field, Proc. Amer. Soc. 12 (1998), 1949-1953. · Zbl 0908.11048 [2] P. J. Cahen and J. L. Chabert, Integers-valued polynomials, Mathematical Surveys and Monographs, Vol. 48, Amer. Math. Soc., Providence, 1997. · Zbl 0884.13010 [3] J. W. S. Cassels and A. Fröhlich, Algebraic Number Theory, Academic Press, 1967. · Zbl 0153.07403 [4] C. Greither and D. K. Harisson, A Galois correspondence for radicals extensions of fields, J. Pure Appl. Algebra 43 (1986), 257-270. · Zbl 0607.12015 [5] B. Heidaryan and A. Rajaei, Pólya fields with only one quadratic Pólya subfield, J. Number Theory 143 (2014), 279-285. · Zbl 1296.11135 [6] B. Heidaryan and A. Rajaei, Some non-Pólya biquadratic fields with low ramification, Rev. Mat. Ibeamericana 33 (2017), 1037-1044. · Zbl 1433.11121 [7] D. Hilbert, Die Theorie der algebraischen Zahlkörper (1897), Jahresbericht der Deutschen Mathematiker-Vereinigung 4 (1894-95), 175-546. · JFM 28.0157.05 [8] J. Hosoya and H. Wada, Tables of ideal class groups of purely cubic fields, Proc. Japan Acad. Ser. A Math. Sci. 68(5) (1992), 111-114. · Zbl 0767.11053 [9] S. Lang, Algebra, 3rd ed., Graduates Texts in Mathematics, Springer, 2002. · Zbl 0984.00001 [10] A. Leriche, Groupes, corps et extensions de Pólya: une question de capitulation, Mathematics, Université de Picardie Jules Verne, 2010. French. <tel-00612597>. [11] A. Leriche, Cubic, quartic and sextic Pólya fields, J. Number Theory 133 (2013), 59-71. · Zbl 1296.11136 [12] P. Llorente and E. Nart, Effective determination of the decomposition of the rational prime in a cubic field, Proc. Amer. Math. Soc. 87(2) (1983), 579-585. · Zbl 0514.12003 [13] J. Neukirch, Algebraic Number Theory, Springer Verlag, Berlin, 1999. · Zbl 0956.11021 [14] A. Ostrowski, Über ganzwertige Polynome in algebraischen Zahlkörpern, J. Reine Angew. Math. 149 (1919), 117-124. · JFM 47.0163.05 [15] G. Pòlya, Über ganzwertige Polynome in algebraischen Zahlkörpern, J. Reine Angew. Math. 149 (1919), 97-116. · JFM 47.0163.04 [16] P. Samuel, Théorie algébrique des nombres, Hermann Collection, Paris Méthodes, 1971. · Zbl 0239.12001 [17] J. P. Serre, Corps locaux, 3e ed., Herman, Paris, 1968. · Zbl 1095.11504 [18] M. Taous, On the Pólya group of some imaginary biquadratic fields, Nonassociative and Non-commutative Algebra and Operator Theory, Springer Proc. Math. Stat., 160, Springer, 2016, pp. 175-182. · Zbl 1366.11108 [19] M. Taous and A. Zekhnini, Pólya groups of the imaginary bicyclic biquadratic number fields, J. Number Theory 177 (2017), 307-327. · Zbl 1428.11185 [20] F. Viviani, Ramification groups and Artin conductors of radical extensions of Q , J. Theorie de Nombres de Bordeaux 16 (2004), 779-816. · Zbl 1075.11073 [21] H. Zantema, Integer valued polynomials over a number fields, Manuscripta Mathematica 40 (1982), 155-203. · Zbl 0505.12003 [22] A. Zekhnini, Imaginary biquadratic Pòlya fields of the form Q(d,−2), Gulf J. Math. 4(4) (2016), 182-188. · Zbl 1405.11143 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.