Nikolaos Marmaridis | Searcharxiv

arXiv2026

Galois Extensions via Finiteness of Orbits

We present an orbit--theoretic reformulation of Galois theory based on the natural action of automorphism groups on fields. Given a field $\mathbf{E}$ and a subgroup $H$ of the automorphism group $\mathrm{Aut}(\mathbf{E})$, we show that algebraic properties of the extension $\mathbf{E}/\mathbf{E}^H$, where $\mathbf{E}^H$ denotes the fixed field of $H$, are encoded in the $H$-orbits arising from the action of $H$ on $\mathbf{E}$. An element $α\in \mathbf{E}$ is algebraic over $\mathbf{E}^H$ if and only if its $H$--orbit is finite. In that case, its minimal polynomial can be explicitly constructed as the product of linear factors over its orbit --a construction that also ensures separability. At the level of field extensions, we prove that $\mathbf{E}/\mathbf{E}^H$ is Galois if and only if all $H$--orbits have finite length, and that $\mathbf{E}/\mathbf{E}^H$ is a finite Galois extension if and only if the lengths of the $H$--orbits are bounded above. This provides a unified orbit--theoretic characterization of algebraicity, separability, normality, and degree. Artin's Lemma is recovered as a direct consequence of this framework. Finally, we show that for simple extensions, the fixed field under a subgroup $H$ of $\mathrm{Aut}(\mathbf{F}(α)/\mathbf{F})$ can be described explicitly by evaluating elementary symmetric polynomials on the $H$--orbit of $α$, provided this orbit is finite. This leads to an effective method for computing fixed fields directly from orbit data. A classical example is included to illustrate the approach.

math.NT↗

arXiv2026

Finite-Orbit Actions and Exact Reconstruction

We associate a profinite group to every group $G$ acting on a set $Ω$ with finite orbits. For each finite $G$-stable subset $A\subseteqΩ$, let $G_A\leq\operatorname{Sym}(A)$ be the induced finite permutation group. The groups $G_A$, with the natural restriction maps, form an inverse system, and we define $Γ_Ω:=\varprojlim_A G_A$. We show that $Γ_Ω$ acts naturally on $Ω$ and is canonically topologically isomorphic to the closure of the image of $G$ in $\operatorname{Sym}(Ω)$, endowed with the topology of pointwise convergence. We introduce the finite-level exactness property $\textup{FLEP}$, under which subgroups of $Γ_Ω$ are recovered up to closure from their fixed-point sets, and closed subgroups are recovered exactly. We prove several equivalent formulations of $\textup{FLEP}$. Under this condition, the fixed-point set construction gives an inclusion-reversing bijection between closed subgroups of $Γ_Ω$ and the fixed subsets of $Ω$ arising from closed subgroups. We apply the theory in two directions. First, every profinite group $Γ$ is recovered from its normal finite-quotient action on $\coprod_{N}Γ/N$, where $N$ ranges over the open normal subgroups of $Γ$. For this action, $\textup{FLEP}$ holds precisely when every finite quotient $Γ/N$, with $N$ open and normal, is a Dedekind group. Second, if $G\leq\Aut(E)$ acts on a field $E$ with finite orbits and $F=E^G$, then $E/F$ is Galois and the construction yields a canonical topological isomorphism $Γ_E \cong_{\mathrm{top}} \operatorname{Gal}(E/F)$, where $\operatorname{Gal}(E/F)$ has the Krull topology. Thus the Krull Galois group is recovered from finite-orbit data.

math.GR↗