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More precisely, for a $K$-regular Galois extension $F|K(t)$, we consider the translates $F(s)|K(s)$ by an extension $K(s)|K(t)$ of rational function fields (in other words, $s$ is a root of $g(X)-t$ for some rational function $g\\in K(X)$). We then show that if $F|K(t)$ is a $K$-regular Galois extension with group $G$ over a number field $K$, then for any degree $k\\ge 2$ and almost all (in a density sense) rational functions $g$ of degree $k$, the translate of $F$ by a root field"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1612.08035","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-12-23T16:59:22Z","cross_cats_sorted":[],"title_canon_sha256":"1fd99be84804d2c8beb8ee3ca398b3b33610b9d9bdb7837c27b87b18b6df8958","abstract_canon_sha256":"ae7878d02ce0b392c187a6f94d18139dc64b3ca5a7e546ae959698db53495aa4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:39.164120Z","signature_b64":"7XeAIpm24DB6+ujm6iWUaLcs5pErI79GqWRYfXZcjzx42R0dxC58wUY8LIYP9+itu6D4E/v5FQspP6LS//XQBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"71dcd028faedcf8d8da23fc55a58a477f158fccc2f2ab3f17b46fc9ef0216224","last_reissued_at":"2026-05-18T00:42:39.163462Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:39.163462Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Non-parametricity of rational translates of regular Galois extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Joachim K\\\"onig","submitted_at":"2016-12-23T16:59:22Z","abstract_excerpt":"We generalize a result of F.\\ Legrand about the existence of non-parametric Galois extensions for a given group $G$. More precisely, for a $K$-regular Galois extension $F|K(t)$, we consider the translates $F(s)|K(s)$ by an extension $K(s)|K(t)$ of rational function fields (in other words, $s$ is a root of $g(X)-t$ for some rational function $g\\in K(X)$). We then show that if $F|K(t)$ is a $K$-regular Galois extension with group $G$ over a number field $K$, then for any degree $k\\ge 2$ and almost all (in a density sense) rational functions $g$ of degree $k$, the translate of $F$ by a root field"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.08035","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1612.08035","created_at":"2026-05-18T00:42:39.163563+00:00"},{"alias_kind":"arxiv_version","alias_value":"1612.08035v2","created_at":"2026-05-18T00:42:39.163563+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.08035","created_at":"2026-05-18T00:42:39.163563+00:00"},{"alias_kind":"pith_short_12","alias_value":"OHONAKH25XHY","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_16","alias_value":"OHONAKH25XHY3DNC","created_at":"2026-05-18T12:30:36.002864+00:00"},{"alias_kind":"pith_short_8","alias_value":"OHONAKH2","created_at":"2026-05-18T12:30:36.002864+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OHONAKH25XHY3DNCH7CVUWFEO7","json":"https://pith.science/pith/OHONAKH25XHY3DNCH7CVUWFEO7.json","graph_json":"https://pith.science/api/pith-number/OHONAKH25XHY3DNCH7CVUWFEO7/graph.json","events_json":"https://pith.science/api/pith-number/OHONAKH25XHY3DNCH7CVUWFEO7/events.json","paper":"https://pith.science/paper/OHONAKH2"},"agent_actions":{"view_html":"https://pith.science/pith/OHONAKH25XHY3DNCH7CVUWFEO7","download_json":"https://pith.science/pith/OHONAKH25XHY3DNCH7CVUWFEO7.json","view_paper":"https://pith.science/paper/OHONAKH2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1612.08035&json=true","fetch_graph":"https://pith.science/api/pith-number/OHONAKH25XHY3DNCH7CVUWFEO7/graph.json","fetch_events":"https://pith.science/api/pith-number/OHONAKH25XHY3DNCH7CVUWFEO7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OHONAKH25XHY3DNCH7CVUWFEO7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OHONAKH25XHY3DNCH7CVUWFEO7/action/storage_attestation","attest_author":"https://pith.science/pith/OHONAKH25XHY3DNCH7CVUWFEO7/action/author_attestation","sign_citation":"https://pith.science/pith/OHONAKH25XHY3DNCH7CVUWFEO7/action/citation_signature","submit_replication":"https://pith.science/pith/OHONAKH25XHY3DNCH7CVUWFEO7/action/replication_record"}},"created_at":"2026-05-18T00:42:39.163563+00:00","updated_at":"2026-05-18T00:42:39.163563+00:00"}