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Denote the absolute Galois group of $K$ by $\\operatorname{Gal}(K)$, and the mod-$n$ central-descending series of $\\operatorname{Gal}(K)$ by $\\operatorname{Gal}(K)^{(i)}$. Recall that Kummer theory, together with our choice of $\\omega$, provides a functorial isomorphism between $\\operatorname{Gal}(K)/\\operatorname{Gal}(K)^{(2)}$ and $\\operatorname{Hom}(K^\\times,\\mathbb{Z}/n)$. Analogously to Kummer theory, in this note we use the Merkurje"},"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":"1310.5613","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-21T15:51:51Z","cross_cats_sorted":[],"title_canon_sha256":"893619aaaba82d1ad865db4fd39f0855ab418806ca0ac042a2fac13374249c02","abstract_canon_sha256":"a6e79973bdaad7521cb72072c954267e0b34486eecb26f61d6e93e2a28698436"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:52.465559Z","signature_b64":"x6YtSHGwHwRDhPmQzkYVOhV7Prz4dE4Q5VXAhaQvnA5GHxuobZQry6XzRCbBGKt/d9L8SufK6BgNZudVDYowBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4ed9b8aea7fa97930488b7d7cbbf7ff8017aa7de2c36c8be64bef1fb7aa06dbc","last_reissued_at":"2026-05-18T02:57:52.464948Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:52.464948Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Abelian-by-Central Galois groups of fields I: a formal description","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Adam Topaz","submitted_at":"2013-10-21T15:51:51Z","abstract_excerpt":"Let $K$ be a field whose characteristic is prime to a fixed integer $n$ with $\\mu_n \\subset K$, and choose $\\omega \\in \\mu_n$ a primitive $n$th root of unity. 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