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Algebraic Geometry for Beginners

Algebraic Geometry for Beginners

          
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About the Book

This book offers a self-contained Elementary introduction to the fundamental concepts and techniques of Algebraic Geometry, leading to some Gems of the subject like Bezout’s Theorem, the Fundamental Theorem of Projective Geometry, and Zariski’s Main Theorem. The book contains a detailed treatment of algebraic plane curves with a special emphasis on elliptic curves and their birational classification. The role played by elliptic curves in modern theory of Cryptology is illustrated. A Novel feature of the book is a discussion of the state of the Art on the Jacobian Problem and its relation to the Epimorphism Theorem. The recently introduced Tame Transformation Method of Cryptosystems, is sketched. Prerequisities are limited to a Knowledge of Finite Galois Theory, and of commutative Noetherian rings. All the Commutative Algebra needed is presented in Chapter 1, and could form the basis for a mini course on the subject. The exposition retains Classroom flavour. About 300 exercises are included, often with adequate hints. Table of Contents 1. COMMUTATIVE Algebra : Nakayama Lemma i. Hilbert Basis Theorem ii. Localisation iii. Graded Rings iv. Dimension Theory v. Integral Extensions vi. Regular and Normal Rings vii. Schmidt and Luroth Theorems viii. Elimination Theory ix. Exercises 2. AFFINE VARIETIES : i. Affine Algebraic Sets ii. Regular Functions iii. Irreducible Algebraic Sets iv. Affine Varieties v. Complete Intersections vi. Examples vii. Morphisms viii. Products ix. Exercises 3. PROJECTIVE VARIETIES : i. Terminology ii. Projective Algebraic Sets iii. Homogenisation/Dehomogenisation iv. Projective Closures v. Morphisms vi. Products vii. Complete Varieties viii. Exercises 4. NON-SINGULAR VARIETIES : i. Tangent Spaces ii. Jacobian Criterion iii. Etale Morphisms iv. Tangent Cones v. Blowing-Up a Point vi. Exercises 5. PLANE CURVES : i. Generalities ii. Rational Curves iii. Multiple Points iv Bezout's Theorem v. Applications of Bezout's Theorem vi. Elliptic Curves and Complex Tori vii. Exercises 6. ZARISKI'S MAIN THEOREM : i. Fibre Dimension of Morphisms ii. Finite Morphisms iii. Normal Varieties iv. Normalisation of Varieties v. Zariki's Main Theorem vi. Jacobian Problem vii. Epimorphism Theorem viii. Exercises


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Product Details
  • ISBN-13: 9788185931661
  • Publisher: Hindustan Book Agency
  • Binding: Paperback
  • Language: English
  • Returnable: Y
  • Weight: 436 gr
  • ISBN-10: 8185931666
  • Publisher Date: 15 Mar 2001
  • Height: 235 mm
  • No of Pages: 354
  • Spine Width: mm
  • Width: 155 mm


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