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We prove some fundamental structure theorems related to these quotients. In particular, it is shown that when $q=p$ is an odd prime, $F_{(3)}$ is the compositum of all Galois extensions $E$ of $F$ such that $\\mathrm{Gal}(E/F)$ is isomorphic to $\\{1\\}$, $\\mathbb{Z}/p$ or to the nonabelian g"},"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":"1103.1508","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-03-08T12:41:12Z","cross_cats_sorted":["math.KT"],"title_canon_sha256":"a422c6ee6f660d965c85e77c0e86bb2280bb835baebc950854a8d216b9df87ce","abstract_canon_sha256":"51e4f06e96c38b551588ecbdfe531634e86a2da7f0333c729b6df76b2df32ff3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:31.872957Z","signature_b64":"78oxqbwhPRhpAo/1tGVNT5s8u/pG+xFjMDwG33C1ZFnzcf9+pX+FxDgBWk85laGQv+6beBFMHipj2Q1Nj7nYAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"961f1fc35f72c91f5d1d0a62cf35e745b42327b8621bab5ddd9e73ba20aa6602","last_reissued_at":"2026-05-18T02:18:31.872450Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:31.872450Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Galois groups and cohomological functors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.KT"],"primary_cat":"math.NT","authors_text":"Ido Efrat, Jan Minac","submitted_at":"2011-03-08T12:41:12Z","abstract_excerpt":"Let $q=p^s$ be a prime power, $F$ a field containing a root of unity of order $q$, and $G_F$ its absolute Galois group. 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