Advanced Algebra Along With A Companion Volume Basic Algebra

Advanced Algebra by Anthony W. Knapp

Advanced Algebra Along With A Companion Volume Basic Algebra by Anthony W. Knapp

Contents of Advanced Algebra Book

  • TRANSITION TO MODERN NUMBER THEORY
  • Historical Background
  • Quadratic Reciprocity
  • Equivalence and Reduction of Quadratic Forms
  • Composition of Forms, Class Group
  • Genera
  • Quadratic Number Fields and Their Units
  • Relationship of Quadratic Forms to Ideals
  • Primes in the Progressions n + and n +
  • Dirichlet Series and Euler Products
  • Dirichlet’s Theorem on Primes in Arithmetic Progressions
  • WEDDERBURN–ARTIN RING THEORY
  • Historical Motivation
  • Semisimple Rings and Wedderburn’s Theorem
  • Rings with Chain Condition and Artin’s Theorem
  • Wedderburn–Artin Radical
  • Wedderburn’s Main Theorem
  • Semisimplicity and Tensor Products
  • Skolem–Noether Theorem
  • Double Centralizer Theorem
  • Wedderburn’s Theorem about Finite Division Rings
  • Frobenius’s Theorem about Division Algebras over the Reals
  • BRAUER GROUP
  • Definition and Examples, Relative Brauer Group
  • Factor Sets
  • Crossed Products
  • Hilbert’s Theorem
  • Digression on Cohomology of Groups
  • Relative Brauer Group when the Galois Group Is Cyclic
  • HOMOLOGICAL ALGEBRA
  • Overview
  • Complexes and Additive Functors
  • Long Exact Sequences
  • Projectives and Injectives
  • Derived Functors
  • Long Exact Sequences of Derived Functors
  • Ext and Tor
  • Abelian Categories
  • THREE THEOREMS IN ALGEBRAIC NUMBER THEORY
  • Setting
  • Discriminant
  • Dedekind Discriminant Theorem
  • Cubic Number Fields as Examples
  • Dirichlet Unit Theorem
  • Finiteness of the Class Number
  • REINTERPRETATION WITH ADELES AND IDELES
  • p-adic Numbers
  • Discrete Valuations
  • Absolute Values
  • Completions
  • Hensel’s Lemma
  • Ramification Indices and Residue Class Degrees
  • Special Features of Galois Extensions
  • Different and Discriminant
  • Global and Local Fields
  • Adeles and Ideles
  • INFINITE FIELD EXTENSIONS
  • Nullstellensatz
  • Transcendence Degree
  • Separable and Purely Inseparable Extensions
  • Krull Dimension
  • Nonsingular and Singular Points
  • Infinite Galois Groups
  • BACKGROUND FOR ALGEBRAIC GEOMETRY
  • Historical Origins and Overview
  • Resultant and Bezout’s Theorem
  • Projective Plane Curves
  • Intersection Multiplicity for a Line with a Curve
  • Intersection Multiplicity for Two Curves
  • General Form of Bezout’s Theorem for Plane Curves
  • Grobner Bases
  • Constructive Existence
  • Uniqueness of Reduced Gr ¨obner Bases
  • Simultaneous Systems of Polynomial Equations
  • THE NUMBER THEORY OF ALGEBRAIC CURVES
  • Historical Origins and Overview
  • Divisors
  • Genus
  • Riemann–Roch Theorem
  • Applications of the Riemann–Roch Theorem
  • METHODS OF ALGEBRAIC GEOMETRY
  • Affine Algebraic Sets and Affine Varieties
  • Geometric Dimension
  • Projective Algebraic Sets and Projective Varieties
  • Rational Functions and Regular Functions
  • Morphisms
  • Rational Maps
  • Zariski’s Theorem about Nonsingular Points
  • Classification Questions about Irreducible Curves
  • Affine Algebraic Sets for Monomial Ideals
  • Hilbert Polynomial in the Affine Case
  • METHODS OF ALGEBRAIC GEOMETRY (Continued)
  • Hilbert Polynomial in the Projective Case
  • Intersections in Projective Space

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