Inflated G-Extensions for Algebraic Number Fields
M Krithika, P Vanchinathan
TL;DR
The paper investigates inflated G-extensions, i.e., finite extensions $K/F$ where $|Aut(K/F)|$ is strictly smaller than the degree $[K:F]$, through the inflation index $[K:F]/|Aut(K/F)|$ and seeks to relate this index to the group $G$ realized as $Aut(K/F)$. Building on Legrand–Paran’s results for Hilbertian fields, it proves a main inflation theorem: if an inflated $G$-extension $K/F$ with inflation index $k\ge1$ exists, then for any $m\ge4$ there are infinitely many inflated $G$-extensions with inflation index $mk$; specialized results for $F=\mathbf{Q}$ yield extra families with index $3k$, and for abelian $G$ there are index $2k$ extensions. The constructions combine linearly disjoint Galois extensions (notably using $A_m$- and $S_m$-Galois groups) with fixed fields to control Aut$(K/F)$, and a disjointness framework with Galois closures to extend inflation to non-Galois base extensions. The work provides concrete method to inflate any realizable $G$-extension to larger indices, enriching the toolkit for the (weaker) inverse Galois problem and highlighting explicit infinite families of inflated extensions over number fields and especially over $\mathbf{Q}$. It thereby connects Hilbertian-field techniques, normalizer computations, and compositum behavior to produce systematically larger inflation indices with prescribed automorphism groups.
Abstract
In 2018, Legrand and Paran proved a weaker form of the Inverse Galois Problem for all Hilbertian fields and all finite groups: that is, there exist possibly non-Galois extensions over given Hilbertian base field with given finite group as the group of field automorphisms fixing the base field. For $\mathbf Q$ it was proved earlier by M. Fried. In this paper our objective is to determine how big the degree of such extension can be compared to the order of the automorphism group. A special case of our result shows that if the Inverse Galois problem for $\bq$ has a solution for a finite group $G$, say of order $n$, then there exist algebraic number fields of degree $nm$, for any $m\ge3$ with the same automorphism group $G$.