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unit; algebraic integer; cubic field; quartic field; quintic field
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 ${\rm Gal}({\mathbb Q}(\varepsilon )/{\mathbb Q})$-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.
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