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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
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