Number theory
CM fields with a reciprocal unit-primitive element
Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 8-12.

Let K be a noncyclotomic CM field. We show that the field KR has a reciprocal unit-primitive element when K does. Also, we prove some related conditions that make the converse of this assertion true.

Soit K un corps CM non cyclotomique. On montre que, si K admet une unité réciproque primitive, il en est de même pour le corps KR. On prouve également des conditions qui rendent vraie l'inverse de cette proposition.

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Accepted:
Published online:
DOI: 10.1016/j.crma.2017.11.014
Greither, Cornelius 1; Zaïmi, Toufik 2

1 Fakultät für Informatik, Institut für theoretische Informatik und Mathematik, Universität der Bundeswehr München, 85577 Neubiberg, Germany
2 Department of Mathematics and Statistics, College of Science, Al Imam Mohammad Ibn Saud Islamic University, Riyadh 11623, Saudi Arabia
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Greither, Cornelius; Zaïmi, Toufik. CM fields with a reciprocal unit-primitive element. Comptes Rendus. Mathématique, Volume 356 (2018) no. 1, pp. 8-12. doi : 10.1016/j.crma.2017.11.014. http://www.numdam.org/articles/10.1016/j.crma.2017.11.014/

[1] C. Batut, D. Bernardi, H. Cohen, M. Olivier, User's Guide to PARI-GP, Version 2.5.1, 2012.

[2] Bertin, M.J.; Zaïmi, T. Complex Pisot numbers in algebraic number fields, C. R. Acad. Sci. Paris, Ser. I, Volume 353 (2015), pp. 965-967

[3] Blanksby, P.E.; Loxton, J.H. A note on the characterization of CM-fields, J. Aust. Math. Soc., Volume 26 (1978), pp. 26-30

[4] Greither, C.; Zaïmi, T. CM fields without unit-primitive elements, Bull. Aust. Math. Soc., Volume 96 (2017), pp. 398-399

[5] Lalande, F. Problèmes de Galois et nombres algébriques réciproques, Université Paris-6, 2000 (thèse de doctorat)

[6] Lang, S. Cyclotomic Fields II, Springer-Verlag, New York, 2012

[7] Zaïmi, T.; Bertin, M.J.; Aljouiee, A. On number fields without unit primitive elements, Bull. Aust. Math. Soc., Volume 93 (2016), pp. 420-432

[8] Zaïmi, T. A note on number fields having reciprocal integer generators, Quaest. Math., Volume 40 (2017), pp. 391-394

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