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They are, for example, the centers of generic algebras with the appropriate kind of involution. This paper considers the rationality properties of these fields, in the case $1,2$ or 4 are the highest powers of 2 that divide $n$. We derive ra"},"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":"math/0102226","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.AG","submitted_at":"2001-02-28T22:39:37Z","cross_cats_sorted":["math.RA"],"title_canon_sha256":"ecc89889a6a031c38ab7fd0c3f4197f0dcd1360db2dff4f7fad593b9d3205828","abstract_canon_sha256":"ba8bfbc05d5497ba91fb5e434b4f4d2b2ee291ff53235d7bd7cb8539954d7367"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:05:37.888489Z","signature_b64":"APw2I/KDltKle1m9RqiBtU68F/e1Lo/nDx5byNnCJ3MTC+xTDP8StgmPRY1PDAAWLf3YfSI1bdTrNrlvjgbWBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2af1b932c924397d2b84ddfb19e638d140a21c30214d0053775ce51e9dd4526f","last_reissued_at":"2026-05-18T01:05:37.888062Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:05:37.888062Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Invariant Fields of Symplectic and Orthogonal Groups","license":"","headline":"","cross_cats":["math.RA"],"primary_cat":"math.AG","authors_text":"David J. Saltman","submitted_at":"2001-02-28T22:39:37Z","abstract_excerpt":"The projective orthogonal and symplectic groups $PO_n(F)$ and $PSp_n(F)$ have a natural action on the $F$ vector space $V' = M_n(F) \\oplus ... \\oplus M_n(F)$. Here we assume $F$ is an infinite field of characteristic not 2. If we assume there is more than one summand in $V'$, then the invariant fields $F(V')^{PO_n}$ and $F(V')^{PSp_n}$ are natural objects. They are, for example, the centers of generic algebras with the appropriate kind of involution. This paper considers the rationality properties of these fields, in the case $1,2$ or 4 are the highest powers of 2 that divide $n$. We derive ra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0102226","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0102226","created_at":"2026-05-18T01:05:37.888137+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0102226v1","created_at":"2026-05-18T01:05:37.888137+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0102226","created_at":"2026-05-18T01:05:37.888137+00:00"},{"alias_kind":"pith_short_12","alias_value":"FLY3SMWJEQ4X","created_at":"2026-05-18T12:25:50.254431+00:00"},{"alias_kind":"pith_short_16","alias_value":"FLY3SMWJEQ4X2K4E","created_at":"2026-05-18T12:25:50.254431+00:00"},{"alias_kind":"pith_short_8","alias_value":"FLY3SMWJ","created_at":"2026-05-18T12:25:50.254431+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FLY3SMWJEQ4X2K4E3X5RTZRY2F","json":"https://pith.science/pith/FLY3SMWJEQ4X2K4E3X5RTZRY2F.json","graph_json":"https://pith.science/api/pith-number/FLY3SMWJEQ4X2K4E3X5RTZRY2F/graph.json","events_json":"https://pith.science/api/pith-number/FLY3SMWJEQ4X2K4E3X5RTZRY2F/events.json","paper":"https://pith.science/paper/FLY3SMWJ"},"agent_actions":{"view_html":"https://pith.science/pith/FLY3SMWJEQ4X2K4E3X5RTZRY2F","download_json":"https://pith.science/pith/FLY3SMWJEQ4X2K4E3X5RTZRY2F.json","view_paper":"https://pith.science/paper/FLY3SMWJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0102226&json=true","fetch_graph":"https://pith.science/api/pith-number/FLY3SMWJEQ4X2K4E3X5RTZRY2F/graph.json","fetch_events":"https://pith.science/api/pith-number/FLY3SMWJEQ4X2K4E3X5RTZRY2F/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FLY3SMWJEQ4X2K4E3X5RTZRY2F/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FLY3SMWJEQ4X2K4E3X5RTZRY2F/action/storage_attestation","attest_author":"https://pith.science/pith/FLY3SMWJEQ4X2K4E3X5RTZRY2F/action/author_attestation","sign_citation":"https://pith.science/pith/FLY3SMWJEQ4X2K4E3X5RTZRY2F/action/citation_signature","submit_replication":"https://pith.science/pith/FLY3SMWJEQ4X2K4E3X5RTZRY2F/action/replication_record"}},"created_at":"2026-05-18T01:05:37.888137+00:00","updated_at":"2026-05-18T01:05:37.888137+00:00"}