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Let O_L be the valuation ring of L and let P_L be its maximal ideal. We show that if L/K is weakly ramified and n is congruent to 1 mod |G_1| then P_L^n is free over the group ring O_K[G], and we construct an explicit generating element. Under the additional assumption that L/K is wildly ramified, we then show that every"},"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":"1204.2133","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-04-10T13:08:19Z","cross_cats_sorted":[],"title_canon_sha256":"5f00307bc70703237b155b65bc31144034a79813c08e0d3789d63f4425b56837","abstract_canon_sha256":"e0bd20e583610c85a70922c6f09c802368eafef00833c8022184719caa60540c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:42:46.820768Z","signature_b64":"UslZhYoYncix/FAj9BcF+qLC+P4n+cTjUYFzkwdNFjgS3jPaI39Tba61vILz53WX+rnn21azPUtuu5wRZUygBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"86a8bfc82a5645fb913e415dcf01cbda20bfa24095ff2f0b632fcb6920e213d5","last_reissued_at":"2026-05-18T02:42:46.820345Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:42:46.820345Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Explicit integral Galois module structure of weakly ramified extensions of local fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Henri Johnston","submitted_at":"2012-04-10T13:08:19Z","abstract_excerpt":"Let L/K be a finite Galois extension of complete local fields with finite residue fields and let G=Gal(L/K). Let G_1 and G_2 be the first and second ramification groups. Thus L/K is tamely ramified when G_1 is trivial and we say that L/K is weakly ramified when G_2 is trivial. Let O_L be the valuation ring of L and let P_L be its maximal ideal. We show that if L/K is weakly ramified and n is congruent to 1 mod |G_1| then P_L^n is free over the group ring O_K[G], and we construct an explicit generating element. Under the additional assumption that L/K is wildly ramified, we then show that every"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.2133","kind":"arxiv","version":4},"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":"1204.2133","created_at":"2026-05-18T02:42:46.820408+00:00"},{"alias_kind":"arxiv_version","alias_value":"1204.2133v4","created_at":"2026-05-18T02:42:46.820408+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1204.2133","created_at":"2026-05-18T02:42:46.820408+00:00"},{"alias_kind":"pith_short_12","alias_value":"Q2UL7SBKKZC7","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_16","alias_value":"Q2UL7SBKKZC7XEJ6","created_at":"2026-05-18T12:27:18.751474+00:00"},{"alias_kind":"pith_short_8","alias_value":"Q2UL7SBK","created_at":"2026-05-18T12:27:18.751474+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/Q2UL7SBKKZC7XEJ6IFO46AOL3I","json":"https://pith.science/pith/Q2UL7SBKKZC7XEJ6IFO46AOL3I.json","graph_json":"https://pith.science/api/pith-number/Q2UL7SBKKZC7XEJ6IFO46AOL3I/graph.json","events_json":"https://pith.science/api/pith-number/Q2UL7SBKKZC7XEJ6IFO46AOL3I/events.json","paper":"https://pith.science/paper/Q2UL7SBK"},"agent_actions":{"view_html":"https://pith.science/pith/Q2UL7SBKKZC7XEJ6IFO46AOL3I","download_json":"https://pith.science/pith/Q2UL7SBKKZC7XEJ6IFO46AOL3I.json","view_paper":"https://pith.science/paper/Q2UL7SBK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1204.2133&json=true","fetch_graph":"https://pith.science/api/pith-number/Q2UL7SBKKZC7XEJ6IFO46AOL3I/graph.json","fetch_events":"https://pith.science/api/pith-number/Q2UL7SBKKZC7XEJ6IFO46AOL3I/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Q2UL7SBKKZC7XEJ6IFO46AOL3I/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Q2UL7SBKKZC7XEJ6IFO46AOL3I/action/storage_attestation","attest_author":"https://pith.science/pith/Q2UL7SBKKZC7XEJ6IFO46AOL3I/action/author_attestation","sign_citation":"https://pith.science/pith/Q2UL7SBKKZC7XEJ6IFO46AOL3I/action/citation_signature","submit_replication":"https://pith.science/pith/Q2UL7SBKKZC7XEJ6IFO46AOL3I/action/replication_record"}},"created_at":"2026-05-18T02:42:46.820408+00:00","updated_at":"2026-05-18T02:42:46.820408+00:00"}