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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.
11R27 Units and factorization
11R16 Cubic and quartic extensions
11R20 Other abelian and metabelian extensions
Full Text: DOI
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