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
• 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’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
• 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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