Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T21:17:27.558Z Has data issue: false hasContentIssue false
  • English
  • Français

ON NUMBER FIELDS WITHOUT A UNIT PRIMITIVE ELEMENT

Part of: Algebraic number theory: global fields

Published online by Cambridge University Press:  11 January 2016

T. ZAÏMI ,
M. J. BERTIN  and
A. M. ALJOUIEE
T. ZAÏMI
Affiliation:
College of Science, Al-Imam Mohammad Ibn Saud Islamic University, PO Box 90950, Riyadh 11623, Saudi Arabia email [email protected]
M. J. BERTIN*
Affiliation:
Université Pierre et Marie Curie (Paris 6), IMJ, 4 Place Jussieu, 75005 Paris, France email [email protected]
A. M. ALJOUIEE
Affiliation:
College of Science, Al-Imam Mohammad Ibn Saud Islamic University, PO Box 90950, Riyadh 11623, Saudi Arabia email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We characterise number fields without a unit primitive element, and we exhibit some families of such fields with low degree. Also, we prove that a noncyclotomic totally complex number field $K$, with degree $2d$ where $d$ is odd, and having a unit primitive element, can be generated by a reciprocal integer if and only if $K$ is not CM and the Galois group of the normal closure of $K$ is contained in the hyperoctahedral group $B_{d}$.

Keywords

primitive elementsPisot numbersunitsCM-fieldsreciprocal integers

MSC classification

Primary: 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Secondary: 11R32: Galois theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 93 , Issue 3 , June 2016 , pp. 420 - 432
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

References

Batut, C., Bernardi, D., Cohen, H. and Olivier, M., User’s Guide to PARI-GP, Version 2.5.1 (Institut de Mathématiques de Bordeaux, 2012), pari.math.u-bordeaux.fr/pub/pari/manuals/2.5.1/users.pdf.Google Scholar
Bertin, M. J., Decomps-Guilloux, A., Grandet-Hugot, M., Pathiaux-Delefosse, M. and Schreiber, J. P., Pisot and Salem numbers (Birkhäuser, Verlag, Basel, 1992).CrossRefGoogle Scholar
Bertin, M. J. and Zaïmi, T., ‘‘Complex Pisot numbers in algebraic number fields’’, C. R. Math. Acad. Sci. Paris 353 (2015), doi:10.1016/j.crma.2015.09.007.Google Scholar
Blanksby, P. E. and Loxton, J. H., ‘A note on the characterization of CM-fields’, J. Aust. Math. Soc. 26 (1978), 26–30.Google Scholar
Bugeaud, Y., Distribution Modulo One and Diophantine Approximation, Cambridge Tracts in Mathematics, 193 (Cambridge University Press, Cambridge, 2012).CrossRefGoogle Scholar
Dubickas, A., ‘Nonreciprocal units in a number field with an application to Oeljeklaus–Toma manifolds (with an appendix by Laurent Battisti)’, New York J. Math. 20 (2014), 257–274.Google Scholar
Edgar, H. M., Mollin, R. A. and Peterson, B. L., ‘Class groups, totally positive units and squares’, Proc. Amer. Math. Soc. 98(1) (1986), 33–37.CrossRefGoogle Scholar
Francisca, C. O., ‘On cyclic cubic fields’, Extracta Math. 6 (1991), 28–30.Google Scholar
Godwin, H. J., ‘The determination of units in totally real cubic fields’, Math. Proc. Cambridge Philos. Soc. 56 (1960), 318–321.Google Scholar
Gras, M. N., ‘Note à propos d’une conjecture de H. J. Godwin sur les unités des corps cubiques’, Ann. Inst. Fourier (Grenoble) 30 (1980), 1–6.Google Scholar
Lalande, F., ‘Corps de nombres engendrés par un nombre de Salem’, Acta Arith. 83 (1999), 191–200.Google Scholar
Miller, V., ‘Two questions about units in number fields’, mathoverflow.net/questions/15260/two-questions-about-units-in-number-fields.Google Scholar
Pisot, C., Quelques aspects de la théorie des entiers algébriques, Séminaire de mathématiques supérieures (Université de Montréal, Montréal, 1966).Google Scholar
SAGE Mathematics Software, Version 3.4, http://www.sagemath.org.Google Scholar
Yokoi, H., ‘On real quadratic fields containing units with norm -1’, Nagoya Math. J. 33 (1968), 139–152.Google Scholar