Title: Selected Topics in Algebra: and its Interrelations with Logic, Number Theory and Algebraic Geometry
Abstract: I. Quadratic Forms and Arithmetical Symbols.- I. Elements of Quadratic Forms Theory Quadratic Forms over a Field k of Characteristic ? 2.- II. Symbols. Elements of the Theory of the Group K2(A).- 1. Symbols.- 2. The Group of the Units of p-adic Fields.- 3. Quadratic Forms over Discrete Valuation Fields.- 4. A Formula for the Hilbert Symbol in the Case of p-adic Fields.- 5. The Product Formula (Hilbert).- 6. Applications. Comments.- 7. The Theorem of Minkowski-Hasse.- II. Methods of Logic and Algorithmic Methods in Algebra.- 1. Countable Sets.- 2. The Notion of Lattice.- 3. The Algebra of Propositions.- 4. Formal Systems.- 5. Primitive Recursive Functions.- 6. Recursive Functions.- 7. Turing Machines.- 8. The Concept of Recursive (Algorithmic) Decidability.- 9. On the Formalist and the Constructive Point of View in Mathematics.- 10. Derived Terms in the Theory of Formal Systems.- 11. The Constructive Approach to Set Theory.- III. Introduction to Modern Algebraic Geometry.- 1. Generalities.- 2. The Spectrum of a Commutative Ring.- 3. Schemes. Chevalley Schemes.- 4. Elements of Projective Geometry.- 5. The Chevalley Scheme Associated to an Integral and Irreducible Scheme.- IV. Theory of Topoi.- I. General Theory of Topoi.- 1. Introduction.- 2. Some Definitions.- 3. Elements of Descent Theory.- 4. Grothendieck Topologies.- II. Theory of Lawvere-Tierney Topoi.- V. Elements of the Theory of Elliptic Curves.- I. Elliptic Curves Defined over the Complex Field.- II. Divisors.- VI. Algebraic Varieties over a Finite Field.- 1. The Zeta Function.- 2. Cohomology Theories.- 3. Weil's Diophantine Conjectures.- VII. Elementary Theory of Non-Standard Real Numbers.- Appendix. On Local Rings with the Approximation Property.
Publication Year: 1984
Publication Date: 1984-09-30
Language: en
Type: book
Access and Citation
AI Researcher Chatbot
Get quick answers to your questions about the article from our AI researcher chatbot