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Using these bijections, we construct a large family of commutative semifields that are non-isotopic to finite fields or twisted fields, which in turn give rise to a large family of non-Desarguesian commutative semifield planes."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","text":"The constructed maps are indeed bijections and the resulting multiplication defines a semifield (i.e., the algebraic identities hold for the chosen parameters), which must be verified explicitly for each member of the family."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","text":"A large family of commutative semifields non-isotopic to fields or Albert twisted fields is obtained from new semiquadratic bijections on P^2(F_q)."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","text":"Semiquadratic homogeneous bijections of the Desarguesian plane produce large families of commutative semifields that are neither fields nor twisted fields."}],"snapshot_sha256":"178242a8d972cdfacbda793c67948ede0c1e5e6f8c0e31b3abd7866906541b97"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The Menichetti-Kaplansky theorem states that a finite semifield that is three-dimensional over its center is either a field or a twisted field of Albert. 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