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It is known that, if $\\bC(G)$ is rational over $\\bC$, then $B_0(G)=0$ where $B_0(G)$ is the unramified Brauer group of $\\bC(G)$ over $\\bC$. Bogomolov showed that, if $G$ is a $p$-group of order $p^5$, then $B_0(G)=0$. This result was disproved by Moravec for $p=3,5,7$ by computer computing. We will give a theoretic proof of the followin"},"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":"1109.2966","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2011-09-14T02:04:37Z","cross_cats_sorted":["math.AG","math.NT"],"title_canon_sha256":"481d329e31b724f2ac8ca384ef263210d07a543e27a901e4ebdb43ec03ff699e","abstract_canon_sha256":"110d1f25c5ca937991b081f1b509972fa23e0dddcf7512accb43d2bee1f837cb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:13:06.615975Z","signature_b64":"w31XDm3RIgXPzQIyLNs8sYQPDVZYto0K6NWA3j38Ig/9J15ca8wOYoElMaHQG1TrlXMf9UM59VjxAJnaM9KMBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"148988e9328aa73dcd0307b971bc7fd5d5f5a909f23c01f1f3c2c0c733df1a43","last_reissued_at":"2026-05-18T04:13:06.615562Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:13:06.615562Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Unramified Brauer groups for groups of order p^5","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.NT"],"primary_cat":"math.AC","authors_text":"Akinari Hoshi, Ming-chang Kang","submitted_at":"2011-09-14T02:04:37Z","abstract_excerpt":"Let $k$ be any field, $G$ be a finite group acting on the rational function field $k(x_g : g\\in G)$ by $h\\cdot x_g=x_{hg}$ for any $h,g\\in G$. Define $k(G)=k(x_g : g\\in G)^G$. Noether's problem asks whether $k(G)$ is rational (= purely transcendental) over $k$. It is known that, if $\\bC(G)$ is rational over $\\bC$, then $B_0(G)=0$ where $B_0(G)$ is the unramified Brauer group of $\\bC(G)$ over $\\bC$. Bogomolov showed that, if $G$ is a $p$-group of order $p^5$, then $B_0(G)=0$. This result was disproved by Moravec for $p=3,5,7$ by computer computing. 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