Institute of Mathematics

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

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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

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