TAILIEUCHUNG - Đề tài " Higher composition laws III: The parametrization of quartic rings "

In the first two articles of this series, we investigated various higher analogues of Gauss composition, and showed how several algebraic objects involving orders in quadratic and cubic fields could be explicitly parametrized. In particular, a central role in the theory was played by the parametrizations of the quadratic and cubic rings themselves. These parametrizations are beautiful and easy to state. | Annals of Mathematics Higher composition laws III The parametrization of quartic rings By Manjul Bhargava Annals of Mathematics 159 2004 1329 1360 Higher composition laws III The parametrization of quartic rings By Manjul Bhargava 1. Introduction In the first two articles of this series we investigated various higher analogues of Gauss composition and showed how several algebraic objects involving orders in quadratic and cubic fields could be explicitly parametrized. In particular a central role in the theory was played by the parametrizations of the quadratic and cubic rings themselves. These parametrizations are beautiful and easy to state. In the quadratic case one need only note that a quadratic ring . any ring that is free of rank 2 as a Z-module is uniquely specified up to isomorphism by its discriminant and conversely given any discriminant D . any integer congruent to 0 or 1 mod 4 there is a unique quadratic ring having discriminant D namely 1 I Z x x2 if D 0 S D Z 1 1 ỰD Z Z if D 1 is a square I Z D y D 2 otherwise. Thus we may say that quadratic rings are parametrized by the set D D E Z D 0 or 1 mod 4 . For a more detailed discussion of quadratic rings see 2 . The cubic case is slightly more complex in that cubic rings are not parametrized only by their discriminants indeed there may sometimes be several cubic orders having the same discriminant. The correct object parametrizing cubic rings . rings free of rank 3 as Z-modules was first determined by Delone-Faddeev in their classic 1964 treatise on cubic irrationalities 8 . They showed that cubic rings are in bijective correspondence with GL2 Z -equivalence classes of integral binary cubic forms as follows. Given a binary cubic form f x y ax3 bx2y cxy2 dy3 with a b c d E Z one associates to f the ring R f having Z-basis 1 w1 w2 and multiplication table 2 A A 2 2 A ad ac bx 1 UX 2 bd dw1 cw2. 1330 MANJUL BHARGAVA One easily verifies that GL2 Z -equivalent binary cubic forms yield isomorphic rings .

• Thạc sĩ - Tiến sĩ - Cao học
• quartic rings
• subjects of mathematics
• Annals of Mathematics
• science reports
• studied mathematics
• natural sciences
• scientific research
• discriminants
• Research methods
• Data collection methods
• Understanding scientific research
• Ensuring research validity
• Measurement techniques
• Sampling methods
• Scientific research capacity
• Educational institutions
• Policy and management studies
• Vietnam Academy for Ethnic minorities
• Scientific bases
• earth sciences
• Research
• VNU Journal of Science
• scientific reports
• Action research
• linguistics
• research projects
• Foreign Languages
• cooperative research
• Renewable energy
• Introduction to research
• Scientific investigation
• Scientific investigation research
• Experimental designs
• Measurement of variables
• Reading research
• Conducting and reading research
• Research process
• Research report
• Purpose of research
• Scientific method
• Conducting research
• The effect of university autonomy on training and scientific research
• Vietnam national university of agriculture
• Training and scientific research
• Vascular plant
• Flora
• Yen Tu
• Quang Ninh Research Forest
• statements and documents scientific reports
• medical reports
• medical knowledge
• medical research
• clinical trials
• medical management
• health care
• Methods in behavioral research
• Behavioral research
• Scientific understanding
• Fundamental research issues
• Measurement concepts
• Observational methods
• reading comprehension
• education
• optimal picket
• model creation
• urban expansion
• organizing spatial
• Journal of Science
• Journal of Technology
• Technology Management
• Research quality
• Strategy and policy
• Summary of Doctoral thesis
• Summary of Doctoral thesis in public Management
• State management
• Female scientific research human
• Research human resource development
• Scientific research and teaching
• Defense and security education
• Current situation and solutions
• Security education
• Structural biochemistry
• epigenetics
• MicroRNAs
• Tạp chí khoa học
• Tạp chí Công nghệ
• Quản lý công nghệ
• Human resources
• Science management
• Poliomyelitis vaccine Research
• Non Ri Maprec
• Production center