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For $1\\le h\\le n$ let $e_h(X_1,\\dots,X_n)$ denote the $h$th elementary symmetric polynomial in $n$ variables, and for $\\alpha\\in L$ set $E_h(\\alpha) =e_h(\\sigma_1(\\alpha),\\dots,\\sigma_n(\\alpha))$. Set $j=\\min\\{v_p(h),\\nu\\}$. We show that for $r\\in\\mathbb{Z}$ we have $E_h(\\mathcal{M}_L^r)\\subset \\mathcal{M}_K^{\\lceil(i_j+hr)/n\\rceil}$, wh"},"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":"1608.07350","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-08-26T02:13:58Z","cross_cats_sorted":[],"title_canon_sha256":"2a0f4a260b9834325a61156f26d7056366e6f55ea9e1307bf86c39078657abd8","abstract_canon_sha256":"66436c949c20acb4997dbb5e33861fbb73f43667f3fc5153c3fcb4609cf7ce5f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:07:53.294333Z","signature_b64":"uz7uAXdKf+CjPnjx4WkZyOwdzFEm2pdxLgqE8pf8g4+MmIrFEjiBRrN/UHPS443W+L+5TD4C9qJ5PX2qsy9NBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b5fbca6d1e5e99ae47fb4bac3fcc3ff045adde72a6993f3abd95dd79f3476da0","last_reissued_at":"2026-05-18T01:07:53.293820Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:07:53.293820Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extensions of local fields and elementary symmetric polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Kevin Keating","submitted_at":"2016-08-26T02:13:58Z","abstract_excerpt":"Let $K$ be a local field whose residue field has characteristic $p$ and let $L/K$ be a finite separable totally ramified extension of degree $n=up^{\\nu}$. Let $\\sigma_1,\\dots,\\sigma_n$ denote the $K$-embeddings of $L$ into a separable closure $K^{sep}$ of $K$. For $1\\le h\\le n$ let $e_h(X_1,\\dots,X_n)$ denote the $h$th elementary symmetric polynomial in $n$ variables, and for $\\alpha\\in L$ set $E_h(\\alpha) =e_h(\\sigma_1(\\alpha),\\dots,\\sigma_n(\\alpha))$. Set $j=\\min\\{v_p(h),\\nu\\}$. We show that for $r\\in\\mathbb{Z}$ we have $E_h(\\mathcal{M}_L^r)\\subset \\mathcal{M}_K^{\\lceil(i_j+hr)/n\\rceil}$, wh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.07350","kind":"arxiv","version":1},"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":"1608.07350","created_at":"2026-05-18T01:07:53.293895+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.07350v1","created_at":"2026-05-18T01:07:53.293895+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.07350","created_at":"2026-05-18T01:07:53.293895+00:00"},{"alias_kind":"pith_short_12","alias_value":"WX54U3I6L2M2","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_16","alias_value":"WX54U3I6L2M24R73","created_at":"2026-05-18T12:30:51.357362+00:00"},{"alias_kind":"pith_short_8","alias_value":"WX54U3I6","created_at":"2026-05-18T12:30:51.357362+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/WX54U3I6L2M24R73JOWD7TB76B","json":"https://pith.science/pith/WX54U3I6L2M24R73JOWD7TB76B.json","graph_json":"https://pith.science/api/pith-number/WX54U3I6L2M24R73JOWD7TB76B/graph.json","events_json":"https://pith.science/api/pith-number/WX54U3I6L2M24R73JOWD7TB76B/events.json","paper":"https://pith.science/paper/WX54U3I6"},"agent_actions":{"view_html":"https://pith.science/pith/WX54U3I6L2M24R73JOWD7TB76B","download_json":"https://pith.science/pith/WX54U3I6L2M24R73JOWD7TB76B.json","view_paper":"https://pith.science/paper/WX54U3I6","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.07350&json=true","fetch_graph":"https://pith.science/api/pith-number/WX54U3I6L2M24R73JOWD7TB76B/graph.json","fetch_events":"https://pith.science/api/pith-number/WX54U3I6L2M24R73JOWD7TB76B/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WX54U3I6L2M24R73JOWD7TB76B/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WX54U3I6L2M24R73JOWD7TB76B/action/storage_attestation","attest_author":"https://pith.science/pith/WX54U3I6L2M24R73JOWD7TB76B/action/author_attestation","sign_citation":"https://pith.science/pith/WX54U3I6L2M24R73JOWD7TB76B/action/citation_signature","submit_replication":"https://pith.science/pith/WX54U3I6L2M24R73JOWD7TB76B/action/replication_record"}},"created_at":"2026-05-18T01:07:53.293895+00:00","updated_at":"2026-05-18T01:07:53.293895+00:00"}