Geodesic
When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures
Czechoslovak Mathematical Journal, Tome 70 (2020) no. 4, pp. 905-919
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.21136/CMJ.2020.0019-19
Classification : 11R16, 11R20, 11R27
Keywords: unit; algebraic integer; cubic field; quartic field; quintic field 
@article{10_21136_CMJ_2020_0019_19,
     author = {Louboutin, St\'ephane R.},
     title = {When is the order generated by a cubic, quartic or quintic algebraic unit {Galois} invariant: three conjectures},
     journal = {Czechoslovak Mathematical Journal},
     pages = {905--919},
     year = {2020},
     volume = {70},
     number = {4},
     doi = {10.21136/CMJ.2020.0019-19},
     mrnumber = {4181786},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0019-19/}
}
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
SP  - 905
EP  - 919
VL  - 70
IS  - 4
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0019-19/
DO  - 10.21136/CMJ.2020.0019-19
LA  - en
ID  - 10_21136_CMJ_2020_0019_19
ER  - 
%0 Journal Article
%A Louboutin, Stéphane R.
%T When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures
%J Czechoslovak Mathematical Journal
%D 2020
%P 905-919
%V 70
%N 4
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2020.0019-19/
%R 10.21136/CMJ.2020.0019-19
%G en
%F 10_21136_CMJ_2020_0019_19
Louboutin, Stéphane R. When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures. Czechoslovak Mathematical Journal, Tome 70 (2020) no. 4, pp. 905-919. doi: 10.21136/CMJ.2020.0019-19

[1] Cohen, H.: A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics 138, Springer, Berlin (1993). | DOI | MR | JFM

[2] Cox, D. A.: Galois Theory. Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts, John Wiley & Sons, Chichester (2004). | DOI | MR | JFM

[3] Kappe, L.-C., Warren, B.: An elementary test for the Galois group of a quartic polynomial. Am. Math. Mon. 96 (1989), 133-137. | DOI | MR | JFM

[4] Lee, J. H., Louboutin, S. R.: On the fundamental units of some cubic orders generated by units. Acta Arith. 165 (2014), 283-299. | DOI | MR | JFM

[5] Lee, J. H., Louboutin, S. R.: Determination of the orders generated by a cyclic cubic unit that are Galois invariant. J. Number Theory 148 (2015), 33-39. | DOI | MR | JFM

[6] Lee, J. H., Louboutin, S. R.: Discriminants of cyclic cubic orders. J. Number Theory 168 (2016), 64-71. | DOI | MR | JFM

[7] Liang, J. J.: On the integral basis of the maximal real subfield of a cyclotomic field. J. Reine Angew. Math. 286/287 (1976), 223-226. | DOI | MR | JFM

[8] Louboutin, S. R.: Hasse unit indices of dihedral octic CM-fields. Math. Nachr. 215 (2000), 107-113. | DOI | MR | JFM

[9] Louboutin, S. R.: Fundamental units for orders generated by a unit. Publ. Math. Besançon, Algèbre et Théorie des Nombres Presses Universitaires de Franche-Comté, Besançon (2015), 41-68. | MR | JFM

[10] Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers. Springer Monographs in Mathematics, Springer, Berlin; PWN-Polish Scientific Publishers, Warszawa (1990). | DOI | MR | JFM

[11] Stevenhagen, P.: Algebra I. Dutch Universiteit Leiden, Technische Universiteit Delft, Leiden, Delft (2017). Available at \brokenlink{ http://websites.math.leidenuniv.nl/algebra/algebra1.{pdf}}

[12] Thaine, F.: On the construction of families of cyclic polynomials whose roots are units. Exp. Math. 17 (2008), 315-331. | DOI | MR | JFM

[13] Thomas, E.: Fundamental units for orders in certain cubic number fields. J. Reine Angew. Math. 310 (1979), 33-55. | DOI | MR | JFM

[14] Yamagata, K., Yamagishi, M.: On the ring of integers of real cyclotomic fields. Proc. Japan Acad., Ser. A 92 (2016), 73-76. | DOI | MR | JFM

Cité par Sources :