Czechoslovak Mathematical Journal, Vol. 70, No. 4, pp. 905-919, 2020
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
