Algebraic Geometry

Algebraic geometry is a branch of mathematics that forges a deep connection between abstract algebra and geometry by studying geometric objects defined by systems of polynomial equations. At its core, it investigates shapes, such as curves and surfaces, not through visual intuition alone, but by analyzing the algebraic properties of the polynomials that describe them, using powerful tools from ring theory and field theory. This interplay allows geometric problems to be translated into algebraic ones and vice versa, making it a central field in pure mathematics with profound applications in number theory, cryptography, and theoretical physics.

1. Foundations of Algebraic Geometry
   1. Historical Context and Motivation
      1. Classical Algebraic Geometry
      2. Modern Developments
      3. Connections to Other Fields
   2. The Algebra-Geometry Dictionary
      1. Basic Philosophy
      2. Correspondence between Ideals and Algebraic Sets
      3. Geometric Interpretation of Algebraic Properties
      4. Functorial Perspective
   3. Prerequisites from Abstract Algebra
      1. Groups
         1. Basic Definitions
         2. Homomorphisms
         3. Quotient Groups
      2. Rings
         1. Definition and Examples
         2. Ring Homomorphisms
         3. Subrings and Ideals
      3. Fields
         1. Definition and Examples
         2. Field Extensions
         3. Algebraic and Transcendental Elements
      4. Vector Spaces
         1. Linear Independence and Bases
         2. Linear Transformations
         3. Dimension Theory