| Title:
|
When is the order generated by a cubic, quartic or quintic algebraic unit Galois invariant: three conjectures (English) |
| Author:
|
Louboutin, Stéphane R. |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
905-919 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 ${\rm Gal}({\mathbb Q}(\varepsilon )/{\mathbb Q})$-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. (English) |
| Keyword:
|
unit |
| Keyword:
|
algebraic integer |
| Keyword:
|
cubic field |
| Keyword:
|
quartic field |
| Keyword:
|
quintic field |
| MSC:
|
11R16 |
| MSC:
|
11R20 |
| MSC:
|
11R27 |
| idZBL:
|
07285969 |
| idMR:
|
MR4181786 |
| DOI:
|
10.21136/CMJ.2020.0019-19 |
| . |
| Date available:
|
2020-11-18T09:41:17Z |
| Last updated:
|
2023-01-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148400 |
| . |
| Reference:
|
[1] Cohen, H.: A Course in Computational Algebraic Number Theory.Graduate Texts in Mathematics 138, Springer, Berlin (1993). Zbl 0786.11071, MR 1228206, 10.1007/978-3-662-02945-9 |
| Reference:
|
[2] Cox, D. A.: Galois Theory.Pure and Applied Mathematics. A Wiley-Interscience Series of Texts, Monographs, and Tracts, John Wiley & Sons, Chichester (2004). Zbl 1057.12002, MR 2119052, 10.1002/9781118033081 |
| Reference:
|
[3] Kappe, L.-C., Warren, B.: An elementary test for the Galois group of a quartic polynomial.Am. Math. Mon. 96 (1989), 133-137. Zbl 0702.11075, MR 0992075, 10.2307/2323198 |
| Reference:
|
[4] Lee, J. H., Louboutin, S. R.: On the fundamental units of some cubic orders generated by units.Acta Arith. 165 (2014), 283-299. Zbl 1307.11120, MR 3263953, 10.4064/aa165-3-7 |
| Reference:
|
[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. Zbl 1394.11073, MR 3283165, 10.1016/j.jnt.2014.09.031 |
| Reference:
|
[6] Lee, J. H., Louboutin, S. R.: Discriminants of cyclic cubic orders.J. Number Theory 168 (2016), 64-71. Zbl 1401.11142, MR 3515806, 10.1016/j.jnt.2016.04.015 |
| Reference:
|
[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. Zbl 0335.12015, MR 0419402, 10.1515/crll.1976.286-287.223 |
| Reference:
|
[8] Louboutin, S. R.: Hasse unit indices of dihedral octic CM-fields.Math. Nachr. 215 (2000), 107-113. Zbl 0972.11105, MR 1768197, 10.1002/1522-2616(200007)215:1<107::aid-mana107>3.0.co;2-a |
| Reference:
|
[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. Zbl 1414.11146, MR 3525537 |
| Reference:
|
[10] Narkiewicz, W.: Elementary and Analytic Theory of Algebraic Numbers.Springer Monographs in Mathematics, Springer, Berlin; PWN-Polish Scientific Publishers, Warszawa (1990). Zbl 0717.11045, MR 1055830, 10.1007/978-3-662-07001-7 |
| Reference:
|
[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}}. |
| Reference:
|
[12] Thaine, F.: On the construction of families of cyclic polynomials whose roots are units.Exp. Math. 17 (2008), 315-331. Zbl 1219.11159, MR 2455703, 10.1080/10586458.2008.10129041 |
| Reference:
|
[13] Thomas, E.: Fundamental units for orders in certain cubic number fields.J. Reine Angew. Math. 310 (1979), 33-55. Zbl 0427.12005, MR 0546663, 10.1515/crll.1979.310.33 |
| Reference:
|
[14] Yamagata, K., Yamagishi, M.: On the ring of integers of real cyclotomic fields.Proc. Japan Acad., Ser. A 92 (2016), 73-76. Zbl 1345.11073, MR 3508577, 10.3792/pjaa.92.73 |
| . |
