When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures

Louboutin, Stéphane R.. "When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures." Czechoslovak Mathematical Journal 70.4 (2020): 905-919. <http://eudml.org/doc/297026>.

@article{Louboutin2020,
abstract = {Let $\varepsilon $ be an algebraic unit of the degree $n\ge 3$. 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.},
author = {Louboutin, Stéphane R.},
journal = {Czechoslovak Mathematical Journal},
keywords = {unit; algebraic integer; cubic field; quartic field; quintic field},
language = {eng},
number = {4},
pages = {905-919},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures},
url = {http://eudml.org/doc/297026},
volume = {70},
year = {2020},
}

TY - JOUR
AU - Louboutin, Stéphane R.
TI - When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures
JO - Czechoslovak Mathematical Journal
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 70
IS - 4
SP - 905
EP - 919
AB - Let $\varepsilon $ be an algebraic unit of the degree $n\ge 3$. 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.
LA - eng
KW - unit; algebraic integer; cubic field; quartic field; quintic field
UR - http://eudml.org/doc/297026
ER -