Algebraic functions and projective curves 1st Edition by David Goldschmidt ISBN 0387954325 9780387954325 by Goldschmidt D. 9780387954325, 0387954325 instant download after payment.

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ISBN 10: 0387954325 
ISBN 13: 9780387954325
Author: David Goldschmidt

Algebraic functions and projective curves 1st Table of contents:

Part I: Algebraic Foundations and Affine Curves

• Chapter 1: Rings, Fields, and Polynomials
  • Basic Ring Theory: Ideals, Prime and Maximal Ideals
  • Field Extensions: Algebraic and Transcendental Extensions
  • Polynomial Rings: Factorization and Irreducibility
  • Resultants and Elimination Theory (Introduction)
• Chapter 2: Affine Algebraic Varieties
  • Affine Space An(k)
  • Affine Varieties: Zeros of Polynomial Ideals
  • The Zariski Topology
  • Irreducible Varieties and Prime Ideals
  • The Nullstellensatz: Bridging Algebra and Geometry
  • Coordinate Rings and Isomorphisms of Varieties
• Chapter 3: Affine Plane Curves
  • Definition and Basic Properties
  • Intersections of Affine Curves (Bézout's Theorem for Affine Curves)
  • Tangent Lines and Singular Points
  • Rational Parametrizations

Part II: Projective Space and Projective Curves

• Chapter 4: Projective Space
  • Construction of Projective Space Pn(k)
  • Homogeneous Coordinates
  • Affine Patches and Homogenization/Dehomogenization
  • Projective Varieties and Ideals
  • The Projective Nullstellensatz
• Chapter 5: Projective Curves: Basic Theory
  • Definition and Properties of Projective Curves
  • Irreducible Components
  • Intersections of Projective Curves: Bézout's Theorem (Full Version)
  • Tangent Lines and Singular Points on Projective Curves
• Chapter 6: Rational Maps and Birational Equivalence
  • Regular Functions and Rational Functions on Varieties
  • Rational Maps and Morphisms
  • Birational Equivalence: Isomorphism in the World of Rational Maps
  • Blow-ups and Resolution of Singularities (Introduction)

Part III: Algebraic Function Fields and Riemann Surfaces

• Chapter 7: Algebraic Function Fields of One Variable
  • Definition and Properties of Function Fields
  • Valuations and Places
  • Discrete Valuation Rings and Local Rings of Curves
  • Relationship between Places and Points on Curves
• Chapter 8: Introduction to Riemann Surfaces
  • Complex Curves and Projective Curves over C
  • Topology of Riemann Surfaces
  • Meromorphic Functions on Riemann Surfaces
  • The Fundamental Connection: Non-Singular Projective Curves and Compact Riemann Surfaces

Part IV: Divisors, Differentials, and the Riemann-Roch Theorem

• Chapter 9: Divisors on Curves
  • Definition of Divisors: Formal Sums of Points
  • Degree of a Divisor
  • Principal Divisors and Linear Equivalence
  • The Group of Divisors and the Picard Group
• Chapter 10: Differentials on Curves
  • Kähler Differentials
  • Canonical Divisor and the Space of Global Differentials
  • Meromorphic Differentials
• Chapter 11: The Riemann-Roch Theorem
  • Statement and Significance of the Theorem
  • Vector Spaces Associated with Divisors (L(D))
  • The Genus of a Curve
  • Applications of Riemann-Roch: Weierstrass Gap Theorem, Embeddings

Part V: Further Topics and Special Curves

• Chapter 12: Elliptic Curves
  • Weierstrass Normal Form
  • The Group Law on Elliptic Curves
  • Torsion Points
  • Applications of Elliptic Curves (Cryptography, Number Theory)
• Chapter 13: Hyperelliptic Curves
  • Definition and Basic Properties
  • The Weierstrass Points
  • Jacobian of a Curve (Brief Introduction)
• Chapter 14: Linear Systems
  • Definition and Properties of Linear Systems
  • Complete Linear Systems
  • Maps Defined by Linear Systems
  • Embeddings of Curves into Projective Space

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Tags: David Goldschmidt, Algebraic, functions