Algebra II: Lecture Notes

Representations
Introduction
What is a Representation?
Examples of Representations
Linear Representations
Characters and The Direct Sum
Review
Characters
Direct Sums
Irreducible Representations
Irreducible Representations
Review
Examples
Invariant Complements
Maschke's Theorem
The Main Theorem
More on Maschke's Theorem
More on Characters
The Main Theorem
Characters and Schur's Lemma
Review
Character Tables
Schur's Lemma
Orthonormality of Characters
Review: Schur's Lemma
An Implication of Schur's Lemma
Matrices and a New Representation
Orthonormality of Characters
Proof of the Main Theorem
Review: Orthonormality of Characters
The Regular Representation
Span of Irreducible Characters
Generalizations to Compact Groups
Rings
What is a Ring?
Zero and Inverses
Homomorphisms
Ideals
Building New Rings
Review
Product Rings
Adjoining Elements to a Ring
Polynomial Rings
Ideals in Polynomial Rings
Ideals in a Field
Polynomial Rings over a Field
Maximal Ideals
Ideals in Multivariate Polynomial Rings
More About Rings
Review: Hilbert's Nullstelensatz
Inverting Elements
Factorization
Factorization in Rings
Review
Euclidean Domains
Polynomial Rings
Greatest Common Divisors
Gauss's Lemma
More Factorization
Factoring Integer Polynomials
Gaussian Primes
Number Fields
The Gaussian Integers
Fermat's Last Theorem, as an Aside
Number Fields
Algebraic Numbers and Integers
Ideal Factorization
Motivation
Prime Ideals
Multiplying Ideals
Lattices
Proof of Unique Factorization
Uniqueness of Ideal Factorization
Properties of Ideal Multiplication
Proof of Unique Factorization
Classification of Prime Ideals
Similarity Classes of Ideals
Ideals in Quadratic Fields
Prime Ideals
The Ideal Class Group
Real Quadratic Number Fields
Function Fields
The Ideal Class Group
Review — Function Fields
Application to Fermat's Last Theorem
Finiteness of the Class Group
Modules over a Ring
Examples
Submodules
Homomorphisms
Generators and Relations
Modules and Presentation Matrices
Review — Definition of Modules
Generators and Relations
Presentation Matrices
Classifying Modules
Elementary Row and Column Operations
Smith Normal Form
Smith Normal Form
Review
Some Examples in Z
Smith Normal Form
Applications
Decomposition of Modules
Classification of Abelian Groups
Uniqueness of Subgroups
The Torsion Subgroup
Polynomial Rings
Noetherian Rings
Noetherian Rings
Submodules over Noetherian Rings
Constructing Noetherian Rings
Hilbert Basis Theorem
Chain Conditions
Fields
Review — Noetherian Rings
Introduction to Fields
Field Extensions
Towers of Extensions
Field Extensions
Primary Fields
Algebraic Elements
Compass and Straightedge Construction
Splitting Fields
Finite Fields
Splitting Fields
Construction of Finite Fields
Structure of Finite Fields
Finite Fields
The Multiplicative Group
Application to Number Theory
Multiple Roots
Geometry of Function Fields
Geometry of Function Fields
Ramified Covers
The Main Theorem of Algebra
The Primitive Element Theorem
Galois Theory
Review: Primitive Element Theorem
The Galois Group
Main Theorem
Examples of Galois Groups
Main Theorem of Galois Theory
Examples of Galois Groups
Proof of Main Theorem
Properties of the Correspondence
Applications of the Galois Correspondence
Review
Cyclotomic Extensions
Kummer Extensions
Quintic Equations
Solving Polynomial Equations
Solvable Groups
Radical Extensions
Symmetric Polynomials
Symmetric Polynomials and the Discriminant
Symmetric Polynomials
The Discriminant
Cubic Polynomials
Solving Polynomial Equations
Cubic Polynomials
Quartic Polynomials
Main Theorem of Algebra
Final Remarks
Galois Theory in Finite Fields
Further Directions
Representation Theory
Compact Lie Groups
Factorization
Rings and Modules
Galois Theory
Dimensions of Irreducible Characters