{"paper":{"title":"Galois Extensions via Finiteness of Orbits","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Nikolaos Marmaridis","submitted_at":"2026-06-30T16:07:26Z","abstract_excerpt":"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 $\\mathbf{E}$.\n  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 c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31900","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.31900/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}