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Algebraic number fields generated by an infinite family of monogenic trinomials

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Abstract

For an infinite family of monogenic trinomials \(P(X)=X^3\pm 3rbX-b\in {\mathbb {Z}}[X]\), arithmetical invariants of the cubic number field \(L={\mathbb {Q}}(\theta )\), generated by a zero \(\theta\) of \(P(X)\), and of its Galois closure \(N=L(\sqrt{d_L})\) are determined. The conductor \(f\) of the cyclic cubic relative extension \(N/K\), where \(K={\mathbb {Q}}(\sqrt{d_L})\) denotes the unique quadratic subfield of \(N\), is proved to be of the form \(3^eb\) with \(e\in \lbrace 1,2\rbrace\), which admits statements concerning primitive ambiguous principal ideals, lattice minima, and independent units in \(L\). The number \(m\) of non-isomorphic cubic fields \(L_1,\ldots ,L_m\) sharing a common discriminant \(d_{L_i}=d_L\) with \(L\) is determined.

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Acknowledgements

The first author acknowledges that his research was supported by the Austrian Science Fund (FWF): projects P26008-N25 and J0497-PHY, and by the Research Executive Agency of the European Union (EUREA). Both authors express their gratitude to the anonymous referee for suggestions improving the layout.

Funding

Research of the first author supported by the Austrian Science Fund (FWF): projects P26008-N25 and J0497-PHY, and by the Research Executive Agency of the European Union (EUREA).

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Authors and Affiliations

  1. Naglergasse 53, 8010, Graz, Austria

    Daniel C. Mayer

  2. Department of Mathematics, Faculty of Sciences and Technology, Moulay Ismail University, Errachidia, Morocco

    Abderazak Soullami

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  1. Daniel C. Mayer
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  2. Abderazak Soullami
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Correspondence to Daniel C. Mayer.

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Dedicated to the memory of Georgi F. Voronoi.

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Mayer, D.C., Soullami, A. Algebraic number fields generated by an infinite family of monogenic trinomials. Bol. Soc. Mat. Mex. 29, 1 (2023). https://doi.org/10.1007/s40590-022-00469-w

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