Galois Extensions via Finiteness of Orbits

Abstract

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 $\alpha \in \mathbf{E}$ 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 $\mathrm{Aut}(\mathbf{F}(\alpha)/\mathbf{F})$ can be described explicitly by evaluating elementary symmetric polynomials on the $H$--orbit of $\alpha$, 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.