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
