{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2006:7WFHLKJMPGYDGB3HSKBAMJLLML","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"aa041f7e51fdb0cbdc4535dfd39e78f6f2f893d943a7f0c47fcf6ac490989263","cross_cats_sorted":["math.AC"],"license":"","primary_cat":"math.NT","submitted_at":"2006-01-23T17:50:43Z","title_canon_sha256":"5b703dd513f3b868bba2932a9b69a1300e95a87a5631e509a0fe3cec15202c4d"},"schema_version":"1.0","source":{"id":"math/0601562","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0601562","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"arxiv_version","alias_value":"math/0601562v2","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0601562","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"pith_short_12","alias_value":"7WFHLKJMPGYD","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"7WFHLKJMPGYDGB3H","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"7WFHLKJM","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:59db2c4cca585a8eb8fb73d1e7439e208b9cea90c49b93dd3a99c44eb250df9e","target":"graph","created_at":"2026-05-18T01:38:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We introduce the universal unitarily graded A-algebra for a commutative ring A and an arbitrary abelian extension U of the group of units of A, and use this concept to give simplified proofs of the main theorems of co-Galois theory in the sense of T. Albu. The main tool is a generalisation of a theorem by M. Kneser which, in our language, is a criterion for the universal algebra to be a field when the base ring A is itself a field. This theorem implies also the theorem of A. Schinzel on linearly independent roots. We discuss examples involving the injective hull of the multiplicative group of ","authors_text":"Almar Kaid, Holger Brenner, Uwe Storch","cross_cats":["math.AC"],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2006-01-23T17:50:43Z","title":"Unitarily graded field extensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0601562","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:dfab0341035cb033fa54de3b9846fe0a5b4dde42950c474792295521363063c3","target":"record","created_at":"2026-05-18T01:38:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"aa041f7e51fdb0cbdc4535dfd39e78f6f2f893d943a7f0c47fcf6ac490989263","cross_cats_sorted":["math.AC"],"license":"","primary_cat":"math.NT","submitted_at":"2006-01-23T17:50:43Z","title_canon_sha256":"5b703dd513f3b868bba2932a9b69a1300e95a87a5631e509a0fe3cec15202c4d"},"schema_version":"1.0","source":{"id":"math/0601562","kind":"arxiv","version":2}},"canonical_sha256":"fd8a75a92c79b0330767928206256b62f33a6021e6a536ed78f2aa726ad53991","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fd8a75a92c79b0330767928206256b62f33a6021e6a536ed78f2aa726ad53991","first_computed_at":"2026-05-18T01:38:24.474653Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:24.474653Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VlNhwYfgO8b3Ctz06tIuJPuM0AmR/FVSf67osKQ5BAJKGvhszVTXGiWbbro4ES+Acnkfa5MXkgkaTXYZMLNaBw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:24.475328Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0601562","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dfab0341035cb033fa54de3b9846fe0a5b4dde42950c474792295521363063c3","sha256:59db2c4cca585a8eb8fb73d1e7439e208b9cea90c49b93dd3a99c44eb250df9e"],"state_sha256":"39eb3df2b7f9a03a22c8b52db186c2a1208635061a1eaa8269ac5e8db0632fd8"}