What the study found
The paper examines how large the degree of an extension can be compared with the order of its automorphism group, where an automorphism group is the set of field symmetries that fix the base field. It states that, in a special case, if the Inverse Galois problem for the rational numbers has a solution for a finite group G of order n, then there are algebraic number fields of degree nm for every m at least 3 with the same automorphism group G.
Why the authors say this matters
The authors place their work in the context of a weaker form of the Inverse Galois Problem, which asks about realizing a given finite group as field automorphisms over a base field. The study suggests that one can control the size of the extension degree relative to the automorphism group in this setting.
What the researchers tested
The paper studies extensions over Hilbertian fields, a class of fields relevant to inverse Galois questions, and compares the degree of such extensions with the size of their automorphism groups. It builds on earlier work by Legrand and Paran, and on an earlier result of M. Fried for the rational numbers.
What worked and what didn't
The abstract says the authors aim to determine how large the extension degree can be compared with the group order. It reports a special case in which the degree can be nm for any m ≥ 3 while the automorphism group remains G.
What to keep in mind
The abstract gives only a special case and does not describe the full theorem or proof details. It also does not state any limitations beyond the scope of the result it reports.
Key points
- The paper studies extensions whose automorphism group is a chosen finite group G.
- It focuses on how the degree of such an extension compares with the order of G.
- A special case says degrees of the form nm are possible for every m ≥ 3 when the group has order n.
- The result is framed in relation to a weaker form of the Inverse Galois Problem for Hilbertian fields.
- The abstract cites earlier work by Legrand and Paran, and an earlier result of M. Fried for Q.
Disclosure
- Research title:
- Degree bounds for extensions with a given automorphism group
- Authors:
- M Krithika, P Vanchinathan
- Publication date:
- 2026-04-24
- OpenAlex record:
- View
Get the weekly research newsletter
Stay current with scholarly research without reading academic papers — one filtered digest, every Friday.