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When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures. (English) Zbl 07285969
Summary: Let $$\varepsilon$$ be an algebraic unit of the degree $$n\geq 3$$. 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 $$\mathrm{Gal}(\mathbb{Q}(\varepsilon)/\mathbb{Q})$$-invariant, i.e. when the $$n$$ complex conjugates $$\varepsilon_1,\dots,\varepsilon_n$$ of $$\varepsilon$$ are in $$\mathbb{Z}[\varepsilon]$$, which amounts to asking that $$\mathbb{Z}[\varepsilon_1,\dots,\varepsilon_n]=\mathbb{Z}[\varepsilon]$$, 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 $$\mathrm{Gal}(\mathbb{Q}(\varepsilon)/\mathbb{Q})$$-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.
##### MSC:
 11R27 Units and factorization 11R16 Cubic and quartic extensions 11R20 Other abelian and metabelian extensions
##### Keywords:
unit; algebraic integer; cubic field; quartic field; quintic field
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