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As in the classical case over Z, the representability of p by the quadratic form X^2+nY^2 is governed by conditions coming from class field theory. A necessary (and almost sufficient) condition is that the ideal generated by p splits completely in the Hilbert class field H of K = F_q(x,sqrt{-n}) (for the appropriate"},"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":"0911.2833","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2009-11-15T06:00:28Z","cross_cats_sorted":[],"title_canon_sha256":"71a9f17033a4c3c432c536deec66d7cb336c664962e3ef063fa80df565a28d08","abstract_canon_sha256":"f9f8b0baf2810fbe8214338d8b251e64840978ab366812199787dd694762e86e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:26:43.770774Z","signature_b64":"wlSD+Hwe7SuNW2xbnF8vxc7Ma+/3Rom28pTrCUGKM5LLBp8FaSkIQviTN8zFtlGVn3oG47T639GQKKse/CkNCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24248a88a4e9aafee47d832f9b2dcba9f79965ab16a5e7b16c34921c7e94d2f0","last_reissued_at":"2026-05-18T04:26:43.769980Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:26:43.769980Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Primes of the form x^2+n*y^2 in function fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Piotr Maciak","submitted_at":"2009-11-15T06:00:28Z","abstract_excerpt":"Let n be a square-free polynomial over F_q, where q is an odd prime power. 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