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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 tmzaemi@imamu.edu.sa
M. J. BERTIN*
Affiliation:
Université Pierre et Marie Curie (Paris 6), IMJ, 4 Place Jussieu, 75005 Paris, France email marie-jose.bertin@imj-prg.fr
A. M. ALJOUIEE
Affiliation:
College of Science, Al-Imam Mohammad Ibn Saud Islamic University, PO Box 90950, Riyadh 11623, Saudi Arabia email amjouiee@imamu.edu.sa
*
✉marie-jose.bertin@imj-prg.fr
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Abstract

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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. 

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