{"paper":{"title":"Skew braces and the Galois correspondence for Hopf Galois structures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"Lindsay N. Childs","submitted_at":"2018-02-09T21:11:13Z","abstract_excerpt":"Let $L/K$ be a Galois extension of fields with Galois group $\\Gamma$, and suppose $L/K$ is also an $H$-Hopf Galois extension. Using the recently uncovered connection between Hopf Galois structures and skew left braces, we introduce a method to quantify the failure of surjectivity of the Galois correspondence from subHopf algebras of $H$ to intermediate subfields of $L/K$, given by the Fundamental Theorem of Hopf Galois Theory. Suppose $L \\otimes_K H = LN$ where $N \\cong (G, \\star)$. Then there exists a skew left brace $(G, \\star, \\circ)$ where $(G, \\circ) \\cong \\Gamma$. We show that there is a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.03448","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}