Article
| Title: | Inflated $G$-extensions for algebraic number fields (English) |
| Author: | Krithika, M. |
| Author: | Vanchinathan, Pichai |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 2 |
| Year: | 2026 |
| Pages: | 565-574 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | In 2018, Legrand and Paran proved a weaker form of the inverse Galois problem: Every finite group appears as the automorphism group of infinitely many finite (possibly non-Galois) extensions of a given Hilbertian base field. For ${\bf Q}$ it was proved earlier by Fried. Our objective is to determine how big the degree of such extension can be when compared to the order of the automorphism group. A special case of our result shows that if the inverse Galois problem for ${\bf Q}$ has a solution for a finite group $G$, say of order $n$, then there exist algebraic number fields of degree $mn$, for any $m\ge 3$ with the same automorphism group $G$. (English) |
| Keyword: | Galois extension |
| Keyword: | inverse Galois problem |
| Keyword: | inflated extension |
| Keyword: | automorphism group |
| MSC: | 11S20 |
| MSC: | 12F12 |
| DOI: | 10.21136/CMJ.2026.0334-25 |
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| Date available: | 2026-05-22T11:22:07Z |
| Last updated: | 2026-05-25 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153649 |
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| Reference: | [1] Fried, E., Kollár, J.: Automorphism groups of algebraic number fields.Math. Z. 163 (1978), 121-123. Zbl 0391.12005, MR 0512465, 10.1007/BF01214058 |
| Reference: | [2] Fried, M.: A note on automorphism groups of algebraic number fields.Proc. Am. Math. Soc. 80 (1980), 386-388. Zbl 0492.12007, MR 0580989, 10.1090/S0002-9939-1980-0580989-8 |
| Reference: | [3] Geyer, W.-D.: Jede endliche Gruppe ist Automorphismengruppe einer endlichen Erweiterung $K|\Bbb{Q}$.Arch. Math. 41 (1983), 139-142 German. Zbl 0515.12005, MR 0719416, 10.1007/BF01196869 |
| Reference: | [4] 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: | [5] Legrand, F., Paran, E.: Automorphism groups over Hilbertian fields.J. Algebra 503 (2018), 1-7. Zbl 1419.12002, MR 3779985, 10.1016/j.jalgebra.2017.12.041 |
| Reference: | [6] Malle, G., Matzat, B. H.: Inverse Galois Theory.Springer Monographs in Mathematics. Springer, Berlin (2018). Zbl 1406.12001, MR 3822366, 10.1007/978-3-662-55420-3 |
| Reference: | [7] 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: | [8] Takahashi, T.: On automorphism groups of global fields.Sugako 32 (1980), 159-160 Japanese. 10.11429/sugaku1947.32.159 |
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