{"id":"W4415374500","doi":"10.4153/s0008439525101343","title":"A classification of double struck upper Q$\\mathbb {Q}$ <mml:math xmlns:xlink=\"http://www.w3.org/1999/xlink\" xmlns:mnf=\"http://cambridge.org/core/manifest\" xmlns:cup=\"http://contentservices.cambridge.org\" xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns:m=\"http://cambridge.org/core/metadata\" xmlns:core=\"http://cambridge.org/core\" xmlns:c=\"http://cambridge.org/core/content\" display=\"inline\"> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> </mml:math> -linear maps from ModifyingAbove caret With bar double struck upper Q times slash ModifyingAbove bar caret With bar double struck upper Q t o r s times$\\overline {\\mathbb {Q}}^\\times /\\overline {\\mathbb {Q}}^\\times _{\\mathrm {tors}}$ <mml:math xmlns:xlink=\"http://www.w3.org/1999/xlink\" xmlns:mnf=\"http://cambridge.org/core/manifest\" xmlns:cup=\"http://contentservices.cambridge.org\" xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns:m=\"http://cambridge.org/core/metadata\" xmlns:core=\"http://cambridge.org/core\" xmlns:c=\"http://cambridge.org/core/content\" display=\"inline\"> <mml:mrow> <mml:mover> <mml:mo>^</mml:mo> <mml:mo stretchy=\"true\">¯</mml:mo> </mml:mover> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> <mml:mo>×</mml:mo> <mml:mo>/</mml:mo> <mml:mover> <mml:mo>_^</mml:mo> <mml:mo stretchy=\"true\">¯</mml:mo> </mml:mover> <mml:mi mathvariant=\"double-struck\">Q</mml:mi> <mml:mstyle mathvariant=\"normal\"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mi>o</mml:mi> <mml:mi>r</mml:mi> <mml:mi>s</mml:mi> </mml:mrow> </mml:mstyle> <mml:mo>×</mml:mo> </mml:mrow> </mml:math> to double struck upper R$\\mathbb {R}$ <mml:math xmlns:xlink=\"http://www.w3.org/1999/xlink\" xmlns:mnf=\"http://cambridge.org/core/manifest\" xmlns:cup=\"http://contentservices.cambridge.org\" xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns:m=\"http://cambridge.org/core/metadata\" xmlns:core=\"http://cambridge.org/core\" xmlns:c=\"http://cambridge.org/core/content\" display=\"inline\"> <mml:mi mathvariant=\"double-struck\">R</mml:mi> </mml:math>","year":2025,"lang":"en","type":"article","venue":"Canadian Mathematical Bulletin","topic":"Advanced Algebra and Geometry","field":"Mathematics","cited_by":0,"is_retracted":false,"has_abstract":true,"ca_institutions":"","funders":"","keywords":"Dual (grammatical number); Space (punctuation); Object (grammar); Representation (politics); Construct (python library); Dual space; Function (biology)","routes":{"ca_aff":false,"ca_fund":false,"ca_venue":true,"about_ca":false,"invisible_to_affiliation_only":true},"retraction":null,"screen":null,"direct_labels":[],"prediction":{"model_version":"metacan-v3-hybrid-931329e0061c","candidate_categories":[],"consensus_categories":[],"category_scores_codex":[0.000740064,0.001452666,0.0007371444,0.003122802,0.002798446,0.005317199,0.001733135,0.001382825,0.06039425],"category_scores_gemma":[0.002755924,0.0009477084,0.001526788,0.002869281,0.003419162,0.005956804,0.004400006,0.003731993,0.03070902],"about_ca_system_candidate":false,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.002282034,"about_ca_system_score_gemma":0.001385408,"about_ca_topic_candidate":false,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.007297425,"about_ca_topic_score_gemma":0.00637672,"domain_scores_codex":[0.9978004,0.0002270081,0.000181428,0.0005652141,0.000757347,0.0004685658],"domain_scores_gemma":[0.9981368,0.0002568535,0.0001358355,0.0005026406,0.0007129004,0.0002549317],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"theoretical_or_conceptual","study_design_scores_codex":[0.0001409477,0.0001493696,0.001953484,0.0002567562,0.00002120452,0.0007226847,0.002367713,0.000784051,0.00924375,0.7432413,0.08353872,0.1575799],"study_design_scores_gemma":[0.00002914836,0.0001271118,0.003314685,0.00013732,0.00003137662,0.00186947,0.001532439,0.002960941,0.01959373,0.1967328,0.7735677,0.0001032431],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":"theoretical_or_conceptual","genre_codex":"other","genre_gemma":"empirical","genre_scores_codex":[0.0446991,0.001689188,0.3846302,0.001564606,0.001683841,0.0002759088,0.00240588,0.007450597,0.5556007],"genre_scores_gemma":[0.3167494,0.002100165,0.1185046,0.001046098,0.0004632845,0.0003799567,0.008834649,0.00727296,0.5446489],"genre_candidate":"empirical","genre_consensus":null,"teacher_disagreement_score":0.06039425,"threshold_uncertainty_score":0.2020389,"prediction_status":"machine_predicted_unvalidated"},"machine_scores":{"provisional":true,"baseline":true,"maturity_gate_passed":false,"score_opus":0.03163640180078299,"score_gpt":0.2532335249600806,"score_spread":0.2215971231592976,"validation_status":"score_only:v0-immature-baseline","note":"Baseline scores from an immature model (maturity gate not passed). 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