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Firstly, we show that if $ L/K $ is a tame Galois extension of $ p $-adic fields then each fractional ideal of $ L $ is free over its associated order in $ H $. We also show that this conclusion remains valid if $ L/K $ is merely almost classically Galois. Finally, we show that if $ L/K $ is an abelian extension of number fields then every ambiguous fractional ideal of $ L"},"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":"1708.06267","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-08-21T14:53:16Z","cross_cats_sorted":[],"title_canon_sha256":"53275afcdb318c96baf5dd254c5288c6076471615c59e1f52906898b89fa2568","abstract_canon_sha256":"19d7deef897e30b6ba9e4695254b72ec9a9f2546226103a50a465c347427126c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:23:14.014919Z","signature_b64":"mo88qZ756YFqRN9sgD/0RFeuIGGpQ1CJxuT1Y8MnoH1sy7cEUF/flpjxDLQOcWIzSe+E3FToZ2A8NHv6AIBEBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1c94c6008216e1a4d183b57a1ad08ef7e07cf1796d83a9651d84291d30877c58","last_reissued_at":"2026-05-18T00:23:14.014289Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:23:14.014289Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Commutative Hopf-Galois module structure of tame extensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Paul J. 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