Czechoslovak Mathematical Journal

Czechoslovak Mathematical Journal, Vol. 76, No. 2, pp. 565-574, 2026

Inflated $G$-extensions for algebraic number fields

M. KRITHIKA, Pichai VANCHINATHAN

Received September 1, 2025. Published online April 24, 2026.

Abstract: 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 $ 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 $ Q$ has a solution for a finite group $G$, say of order $n$, then there exist algebraic number fields of degree $mn$, for any $m\ge3$ with the same automorphism group $G$.

Keywords: Galois extension; inverse Galois problem; inflated extension; automorphism group

Classification MSC: 11S20, 12F12

DOI: 10.21136/CMJ.2026.0334-25

PDF available at: Springer Institute of Mathematics CAS Digital Mathematics Library

References:
[1] E. Fried, J. Kollár: Automorphism groups of algebraic number fields. Math. Z. 163 (1978), 121-123. DOI 10.1007/BF01214058 | MR 0512465 | Zbl 0391.12005
[2] M. Fried: A note on automorphism groups of algebraic number fields. Proc. Am. Math. Soc. 80 (1980), 386-388. DOI 10.1090/S0002-9939-1980-0580989-8 | MR 0580989 | Zbl 0492.12007
[3] W.-D. Geyer: Jede endliche Gruppe ist Automorphismengruppe einer endlichen Erweiterung $K|\Bbb{Q}$. Arch. Math. 41 (1983), 139-142. (In German.) DOI 10.1007/BF01196869 | MR 0719416 | Zbl 0515.12005
[4] M. Krithika, P. Vanchinathan: 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. DOI 10.1007/s12044-024-00799-x | MR 4796808 | Zbl 1559.11125
[5] F. Legrand, E. Paran: Automorphism groups over Hilbertian fields. J. Algebra 503 (2018), 1-7. DOI 10.1016/j.jalgebra.2017.12.041 | MR 3779985 | Zbl 1419.12002
[6] G. Malle, B. H. Matzat: Inverse Galois Theory. Springer Monographs in Mathematics. Springer, Berlin (2018). DOI 10.1007/978-3-662-55420-3 | MR 3822366 | Zbl 1406.12001
[7] A. R. Perlis: Roots appear in quanta. Am. Math. Mon. 111 (2004), 61-63. DOI 10.1080/00029890.2004.11920051 | MR 2026318 | Zbl 1076.12003
[8] T. Takahashi: On automorphism groups of global fields. Sugako 32 (1980), 159-160. (In Japanese.) DOI 10.11429/sugaku1947.32.159

Affiliations: M. Krithika, Vellore Institute of Technology University, Vandalur Kelambakkam Road, Keelakotaiyur, Chennai, Tamil Nadu - 600 127, India, e-mail: krithika.m2020@vitstudent.ac.in; Pichai Vanchinathan (corresponding author), H4C 383 Tambaram Road, Chennai, 600042, India, e-mail: vanchinathan@gmail.com

Springer subscribers can access the papers on Springer website.

[List of online first articles] [Contents of Czechoslovak Mathematical Journal] [Full text of the older issues of Czechoslovak Mathematical Journal at DML-CZ]

PDF available at:

Springer
for subscribers

···

Institute of Mathematics CAS
fulltext not available (moving wall 24 months)

···

Digital Mathematics Library
fulltext not available (moving wall 24 months)