Czechoslovak Mathematical Journal, first online, pp. 1-15


When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures

Stéphane R. Louboutin

Received January 15, 2019.   Published online March 30, 2020.

Abstract:  Let $\varepsilon$ be an algebraic unit of the degree $n\geq3$. Assume that the extension ${\mathbb Q}(\varepsilon)/{\mathbb Q}$ is Galois. We would like to determine when the order ${\mathbb Z}[\varepsilon]$ of ${\mathbb Q}(\varepsilon)$ is ${\rm Gal}({\mathbb Q}(\varepsilon)/{\mathbb Q})$-invariant, i.e. when the $n$ complex conjugates $\varepsilon_1,\cdots,\varepsilon_n$ of $\varepsilon$ are in ${\mathbb Z}[\varepsilon]$, which amounts to asking that ${\mathbb Z}[\varepsilon_1,\cdots,\varepsilon_n]={\mathbb Z}[\varepsilon]$, i.e., that these two orders of ${\mathbb Q}(\varepsilon)$ have the same discriminant. This problem has been solved only for $n=3$ by using an explicit formula for the discriminant of the order ${\mathbb Z}[\varepsilon_1,\varepsilon_2,\varepsilon_3]$. However, there is no known similar formula for $n>3$. In the present paper, we put forward and motivate three conjectures for the solution to this determination for $n=4$ (two possible Galois groups) and $n=5$ (one possible Galois group). In particular, we conjecture that there are only finitely many cyclic quartic and quintic Galois-invariant orders generated by an algebraic unit. As a consequence of our work, we found a parametrized family of monic quartic polynomials in ${\mathbb Z}[X]$ whose roots $\varepsilon$ generate bicyclic biquadratic extensions ${\mathbb Q}(\varepsilon)/{\mathbb Q}$ for which the order ${\mathbb Z}[\varepsilon]$ is ${\rm Gal}({\mathbb Q}(\varepsilon)/{\mathbb Q})$-invariant and for which a system of fundamental units of ${\mathbb Z}[\varepsilon]$ is known. According to the present work it should be difficult to find other similar families than this one and the family of the simplest cubic fields.
Keywords:  unit; algebraic integer; cubic field; quartic field; quintic field
Classification MSC:  11R27, 11R16, 11R20
DOI:  10.21136/CMJ.2020.0019-19

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] H. Cohen: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics 138, Springer, Berlin (1993). DOI 10.1007/978-3-662-02945-9 | MR 1228206 | Zbl 0786.11071
[2] D. A. Cox: Galois Theory. Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts, John Wiley & Sons, Chichester (2004). DOI 10.1002/9781118033081 | MR 2119052 | Zbl 1057.12002
[3] L.-C. Kappe, B. Warren: An elementary test for the Galois group of a quartic polynomial. Am. Math. Mon. 96 (1989), 133-137. DOI 10.2307/2323198 | MR 0992075 | Zbl 0702.11075
[4] J. H. Lee, S. R. Louboutin: On the fundamental units of some cubic orders generated by units. Acta Arith. 165 (2014), 283-299. DOI 10.4064/aa165-3-7 | MR 3263953 | Zbl 1307.11120
[5] J. H. Lee, S. R. Louboutin: Determination of the orders generated by a cyclic cubic unit that are Galois invariant. J. Number Theory 148 (2015), 33-39. DOI 10.1016/j.jnt.2014.09.031 | MR 3283165 | Zbl 1394.11073
[6] J. H. Lee, S. R. Louboutin: Discriminants of cyclic cubic orders. J. Number Theory 168 (2016), 64-71. DOI 10.1016/j.jnt.2016.04.015 | MR 3515806 | Zbl 1401.11142
[7] J. J. Liang: On the integral basis of the maximal real subfield of a cyclotomic field. J. Reine Angew. Math. 286/287 (1976), 223-226. DOI 10.1515/crll.1976.286-287.223 | MR 0419402 | Zbl 0335.12015
[8] S. R. Louboutin: Hasse unit indices of dihedral octic CM-fields. Math. Nachr. 215 (2000), 107-113. DOI 10.1002/1522-2616(200007)215:1<107::aid-mana107>3.0.co;2-a | MR 1768197 | Zbl 0972.11105
[9] S. R. Louboutin: Fundamental units for orders generated by a unit. Publ. Math. Besançon, Algèbre et Théorie des Nombres Presses Universitaires de Franche-Comté, Besançon (2015), 41-68. MR 3525537 | Zbl 1414.11146
[10] W. Narkiewicz: Elementary and Analytic Theory of Algebraic Numbers. Springer Monographs in Mathematics, Springer, Berlin; PWN-Polish Scientific Publishers, Warszawa (1990). DOI 10.1007/978-3-662-07001-7 | MR 1055830 | Zbl 0717.11045
[11] P. Stevenhagen: Algebra I. Dutch Universiteit Leiden, Technische Universiteit Delft, Leiden, Delft (2017). Available at http://websites.math.leidenuniv.nl/algebra/algebra1.pdf.
[12] F. Thaine: On the construction of families of cyclic polynomials whose roots are units. Exp. Math. 17 (2008), 315-331. DOI 10.1080/10586458.2008.10129041 | MR 2455703 | Zbl 1219.11159
[13] E. Thomas: Fundamental units for orders in certain cubic number fields. J. Reine Angew. Math. 310 (1979), 33-55. DOI 10.1515/crll.1979.310.33 | MR 0546663 | Zbl 0427.12005
[14] K. Yamagata, M. Yamagishi: On the ring of integers of real cyclotomic fields. Proc. Japan Acad., Ser. A 92 (2016), 73-76. DOI 10.3792/pjaa.92.73 | MR 3508577 | Zbl 1345.11073

Affiliations:   Stéphane R. Louboutin, Aix Marseille Université, CNRS, Centrale Marseille, I2M, 39, rue Frédéric Joliot-Curie, 13453 Marseille Cedex 13, France, e-mail: stephane.louboutin@univ-amu.fr


 
PDF available at: