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

Czechoslovak Mathematical Journal (2020)

- Volume: 70, Issue: 4, page 905-919
- ISSN: 0011-4642

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topLouboutin, 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 -

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