{"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":"codex-gemma-dda1882f352a","candidate_categories":["metaepi_narrow","metaepi_broad","sts","scholarly_communication","open_science","research_integrity","insufficient_payload"],"consensus_categories":["metaepi_narrow","sts","scholarly_communication","open_science","research_integrity","insufficient_payload"],"category_scores_codex":[0.01436174,0.01541519,0.00877092,0.00690449,0.01304337,0.01151471,0.02287592,0.01898134,0.008841092],"category_scores_gemma":[0.008155437,0.02146995,0.01578322,0.01203235,0.01232346,0.01556424,0.01872506,0.02081571,0.02383673],"about_ca_system_candidate":true,"about_ca_system_consensus":false,"about_ca_system_score_codex":0.00133069,"about_ca_system_score_gemma":0.01421323,"about_ca_topic_candidate":true,"about_ca_topic_consensus":false,"about_ca_topic_score_codex":0.03349135,"about_ca_topic_score_gemma":0.007188333,"domain_scores_codex":[0.9158471,0.002577539,0.02136123,0.0195186,0.019788,0.02090748],"domain_scores_gemma":[0.9316702,0.008243301,0.02255053,0.01994815,0.005031044,0.01255681],"domain_codex":null,"domain_gemma":null,"domain_candidate":null,"domain_consensus":null,"study_design_codex":"theoretical_or_conceptual","study_design_gemma":"simulation_or_modeling","study_design_scores_codex":[0.02090126,0.00527173,0.001216137,0.01149335,0.01783094,0.01129318,0.003111046,0.004894139,0.01383317,0.8608998,0.04776445,0.001490797],"study_design_scores_gemma":[0.03899075,0.01162756,0.003772644,0.01774588,0.0253968,0.01673957,0.0168781,0.6393339,0.1746627,0.002354831,0.02797131,0.02452593],"study_design_candidate":"theoretical_or_conceptual","study_design_consensus":null,"genre_codex":"empirical","genre_gemma":"empirical","genre_scores_codex":[0.9164321,0.006996549,0.00308537,0.002891571,0.01502175,0.002885279,0.0170987,0.004511546,0.03107711],"genre_scores_gemma":[0.8987874,0.007791396,0.01272049,0.009011793,0.01435493,0.01257658,0.02962716,0.009652502,0.005477713],"genre_candidate":"empirical","genre_consensus":"empirical","teacher_disagreement_score":0.8585449,"threshold_uncertainty_score":0.9982046,"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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