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We show that if S/R is totally ramified and its degree n is a power of p, then any element $\\rho$ of L with $v_L(\\rho)$ congruent to $-v_L(D_{S/R})-1$ mod n generates L as an H-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. Elder for Galois extensions."},"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":"0905.3085","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-05-19T12:40:53Z","cross_cats_sorted":[],"title_canon_sha256":"1ea3f394826d43b0178ec8b325d8b9858f08b83d5941eac67df78bb40e696a3a","abstract_canon_sha256":"37f069e613a611b5d872ff2d8a7571512e4c4d1cc5ad3fa9151a0bafd7c40e00"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:30:02.966616Z","signature_b64":"HmBPMLnBVQXLtp9LImHmvahphwPJuKvlFmqerLGWLeT+OdvNz7L7CeFj077uYncQJEiBkim1amW1cDB6LAteDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3a893581a807cc474f5034fafef12f5c8ec3709f8cf79bcdc9445e5b6e7f0204","last_reissued_at":"2026-05-18T04:30:02.966000Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:30:02.966000Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Valuation Criterion for Normal Basis Generators of Hopf-Galois Extensions in Characteristic p","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Nigel P. Byott","submitted_at":"2009-05-19T12:40:53Z","abstract_excerpt":"Let S/R be a finite extension of discrete valuation rings of characteristic p>0, and suppose that the corresponding extension L/K of fields of fractions is separable and is H-Galois for some K-Hopf algebra H. Let D_{S/R} be the different of S/R. We show that if S/R is totally ramified and its degree n is a power of p, then any element $\\rho$ of L with $v_L(\\rho)$ congruent to $-v_L(D_{S/R})-1$ mod n generates L as an H-module. This criterion is best possible. These results generalise to the Hopf-Galois situation recent work of G. 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