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Let $N/K$ be a purely inseparable field extension. For the field extensions $L/K$ and $NL/N$, the aim of the paper is to give explicit descriptions of the following subfields and their degrees in terms of the coefficients of the polynomial $f$ and two numerical field invariants $m_f$ and $m_{f,N}$: $L^{pi}$, $L^{pi}L^{sep}$, $(NL)^{pi}$ and $(NL)^{pi}(NL)^{sep}$. From these results, we derive new expl"},"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":"2606.19962","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.RA","submitted_at":"2026-06-18T08:59:32Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"9639cd19b6d408975fd27077e289d8fef7f67093f9d46a918f161d5dde8067c5","abstract_canon_sha256":"0c4744f66eaeca0ac0efbff9dcf7a1b050e59e48702f203b4c113cf3802fbe29"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-19T16:12:59.278547Z","signature_b64":"YsE817d/U0IYT7amcYgVT8v+O5iJ/BR9/OeMnUNZanUsb+60Ulp3zQr4GWkJsHTYjN2G2v2rqSv6P9QEKqEMAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"656af25e2047e8e8f00c5066aaad7841bd1c20c8b219f5ab810485d1d7cc3f30","last_reissued_at":"2026-06-19T16:12:59.278197Z","signature_status":"signed_v1","first_computed_at":"2026-06-19T16:12:59.278197Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Explicit descriptions of the subfields $(NL)^{pi}$ and $(NL)^{pi}(NL)^{sep}$ of $NL$ and new explicit criteria for $NL = (NL)^{pi}(NL)^{sep}$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.RA","authors_text":"V. V. Bavula","submitted_at":"2026-06-18T08:59:32Z","abstract_excerpt":"Let $L=K(\\theta)\\simeq K[x]/f(x)$ be a simple field extension in prime characteristic $p>0$, $L^{sep}$ and $L^{pi}$ be the maximal separable and purely inseparable subfields of $L$, respectively. Let $N/K$ be a purely inseparable field extension. For the field extensions $L/K$ and $NL/N$, the aim of the paper is to give explicit descriptions of the following subfields and their degrees in terms of the coefficients of the polynomial $f$ and two numerical field invariants $m_f$ and $m_{f,N}$: $L^{pi}$, $L^{pi}L^{sep}$, $(NL)^{pi}$ and $(NL)^{pi}(NL)^{sep}$. 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