Article
| Title: | On minimal generating sets of splitting field, cluster towers and multiple transitivity of Galois groups (English) |
| Author: | Jaiswal, Shubham |
| Author: | Vanchinathan, Pichai |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 3 |
| Year: | 2026 |
| Pages: | 1009-1027 |
| Summary lang: | English |
| . | |
| Category: | math |
| . | |
| Summary: | A natural generating set for a Galois extension regarded as the splitting field of an irreducible polynomial is introduced and investigated here. Minimal generating sets arising in this context throw many surprises compared to the analogous concept in the context of vector spaces: they can be of different cardinalities. In fact, we establish that for a certain family of polynomials over the rationals, we have minimal generating sets of all cardinalities in a certain range and that these are the only possible cardinalities for minimal generating set for such a polynomial. We also study how minimal generating sets behave under multiple transitivity of the Galois group and consequently prove the existence of polynomials with all minimal generating sets of uniformly same cardinality. We also connect minimal generating sets with the concept of root cluster tower of an irreducible polynomial introduced in M. Krithika, P. Vanchinathan (2024). (English) |
| Keyword: | minimal generating set of splitting field of polynomial |
| Keyword: | cluster tower |
| Keyword: | multiple transitivity of Galois group |
| MSC: | 11R32 |
| MSC: | 12F05 |
| MSC: | 12F10 |
| MSC: | 20B20 |
| MSC: | 20B35 |
| DOI: | 10.21136/CMJ.2026.0018-26 |
| . | |
| Date available: | 2026-09-11T13:17:11Z |
| Last updated: | 2026-09-14 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153740 |
| . | |
| Reference: | [1] Arias-de-Reyna, S., König, J.: Locally cyclic extensions with Galois group $GL_2(p)$.Int. J. Number Theory 20 (2024), 781-796. Zbl 1557.12004, MR 4736376, 10.1142/S1793042124500398 |
| Reference: | [2] Bhagwat, C., Jaiswal, S.: Cluster magnification, root capacity, unique chains, base change and ascending index.Proc. Indian Acad. Sci., Math. Sci. 135 (2025), Article ID 19, 34 pages. Zbl 1573.11347, MR 4927994, 10.1007/s12044-025-00823-8 |
| Reference: | [3] Brink, D.: On alternating and symmetric groups as Galois groups.Isr. J. Math. 142 (2004), 47-60. Zbl 1067.12004, MR 2085710, 10.1007/BF02771527 |
| Reference: | [4] Dixon, J. D., Mortimer, B.: Permutation Groups.Graduate Texts in Mathematics 163. Springer, New York (1996). Zbl 0951.20001, MR 1409812, 10.1007/978-1-4612-0731-3 |
| Reference: | [5] Edwards, H. M.: Galois Theory.Graduate Texts in Mathematics 101. Springer, New York (1984). Zbl 0532.12001, MR 0743418 |
| Reference: | [6] Gómez-Molleda, M. A., Lario, J.-C.: Sharply $k$-transitive permutation groups viewed as Galois groups.Mediterr. J. Math. 8 (2011), 617-632. Zbl 1257.12004, MR 2860689, 10.1007/s00009-010-0100-x |
| Reference: | [7] Jacobson, E. T., Vélez, W. Y.: The Galois group of a radical extension of the rationals.Manuscr. Math. 67 (1990), 271-284. Zbl 0717.12002, MR 1046989, 10.1007/BF02568433 |
| Reference: | [8] Krithika, M., Vanchinathan, P.: An elementary problem in Galois theory about the roots of irreducible polynomials.Proc. Indian Acad. Sci., Math. Sci. 134 (2024), Article ID 28, 9 pages. Zbl 1559.11125, MR 4796808, 10.1007/s12044-024-00799-x |
| Reference: | [9] Malle, G., Matzat, B. H.: Inverse Galois Theory.Springer Monographs in Mathematics. Springer, Berlin (1999). Zbl 0940.12001, MR 1711577, 10.1007/978-3-662-12123-8 |
| Reference: | [10] McKay, J., Stauduhar, R.: Finding relations among the roots of an irreducible polynomial.Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation ACM Press, New York (1997), 75-77. Zbl 0942.12002, MR 1809972, 10.1145/258726.258752 |
| Reference: | [11] Morandi, P.: Field and Galois Theory.Graduate Texts in Mathematics 167. Springer, New York (1996),\99999DOI99999 10.1007/978-1-4612-4040-2 \catcode` =11. Zbl 0865.12001, MR 1410264 |
| Reference: | [12] Perlis, A. R.: Roots appear in quanta: Exercise solutions.Available at \brokenlink{https://www.{math.lsu.edu/ aperlis/publications/rootsinquanta/}} (2003), 4 pages \catcode` =13. |
| Reference: | [13] Perlis, A. R.: Roots appear in quanta.Am. Math. Mon. 111 (2004), 61-63. Zbl 1076.12003, MR 2026318, 10.1080/00029890.2004.11920051 |
| Reference: | [14] Shafarevich, I. R.: Factors of decreasing central series.Math. Notes 45 (1989), 262-264. Zbl 0681.20024, MR 1001703, 10.1007/BF01158564 |
| Reference: | [15] Völklein, H.: Groups as Galois Groups: An Introduction.Cambridge Studies in Advanced Mathematics 53. Cambridge University Press, Cambridge (1996). Zbl 0868.12003, MR 1405612, 10.1017/cbo9780511471117 |
| . |
Fulltext not available (moving wall 24 months)
Search
Partner of
