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Modern Algebra:
Groups, Rings, Modules, Fields
Sam Macdonald
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You can't use 'macro parameter character #' in math mode
You can't use 'macro parameter character #' in math mode
Front Matter
Colophon
Acknowledgements
How to Use This Book
The Structure of this Text
I
Group Theory
1
Intro to Groups
1.1
Group Basics
Binary Operations and the Definition of a Group
First Properties of Groups
Abelian Groups
First Examples of Groups
Groups of Units
Other Initial Examples and Properties
1.2
Order and Power
Power
Order
1.3
Permutations and Symmetric Groups
Symmetric Groups
Cycles and Transpositions
Even and Odd Permutations
1.4
Dihedral Groups
Dihedral Groups
1.5
Group Homomorphisms
Homomorphism? I Barely Know ’em!
Isomorphism? I Know ’em!
Automorphism? I’m
Am
’em!
1.6
Subgroups
Subgroups
The Subgroup Lattice
Stuck in the Middle
2
Generators, Cyclic Groups
2.1
Generated Subgroups
Generational Wealth
2.2
Cyclic Groups
Definition and First Properties
Subgroups of Cyclic Groups
Uniqueness of Cyclic Groups
3
Normal Subgroups, Quotient Groups
3.1
Cosets and Lagrange’s Theorem
Cosets
The Index of a Subgroup
3.2
Normal Subgroups
The New Normal
3.3
Quotient Groups
Quotient Groups
3.4
Products and the Isomorphism Theorems
The First Isomorphism Theorem
Products
4
Presentations, Free Groups (*)
4.1
Free Groups
4.2
Presentations
What is a Presentation?
Every Group is a Quotient of a Free Group
5
Group Actions
5.1
Group Actions
Time to Act
The Permutation Representation
Faithful and Transitive Actions
5.2
Self Actions and Conjugacy
Self Actions
Conjugacy
5.3
Orbits and Stabilizers
In a Stable Orbit
The Class Equation
6
Simple, Solvable, Sylow
6.1
Sylow’s Theorems
Groups of Prime Order
Sylow Subgroups
Sylows-Theorem
6.2
Simple Groups
Keeping it Simple
A Wealth of Problems on Simple Groups
6.3
Solvable Groups (*)
Composition Series
Solvable Groups
7
Group Products
7.1
Direct Products
Direct Products of Groups
Internal and External Direct Products
7.2
Semidirect Products
External Semidirect Products
Internal Semidirect Products
Groups of Order
p
q
7.3
Finitely Generated Abelian Groups
II
Ring Theory
8
Intro to Rings
8.1
Rings, Subrings
From Rings to Fields
Subrings
8.2
Units, Zerodivisors, Integral Domains
Absolute Units
A Zero Divided Cannot Stand
8.3
Group Rings, Polynomial Rings
What is a Group Ring?
What is a Polynomial Ring?
8.4
Homomorphisms, Polynomial Rings
Homomorphisms
Polynomial Rings
9
Ideals and Quotient Rings
9.1
Ideals
Ideal or No Deal
Generated Ideals
9.2
Quotient Rings, the Ring Isomorphism Theorems
Quotient Rings
The Ring Isomorphism Theorems
9.3
Prime and Maximal Ideals
Prime Ideals
Maximal Ideals
10
Domains
10.1
Euclidean Domains (EDs)
10.2
Principal Ideal Domains (PIDs)
Principal Ideals
Principal Ideal Domains
10.3
Unique Factorization Domains (UFDs)
10.4
Noetherian Rings
11
Fields of Fractions, Localization
11.1
Fields of Fractions
III
Module Theory
12
Intro to Modules
12.1
Modules and Algebras
Modules
Restriction of Scalars and Algebras
12.2
Module Homomorphisms
Homomorphisms and Isomorphisms
Hom
Quotient Modules
13
Bases, Free Modules, Vector Spaces
13.1
Generated Modules and Linear Independence
Generated Modules
Linear Independence
13.2
Free Modules
Bases
Rank
13.3
Vector Spaces
Every Vector Space has a Basis
Every Basis of a Vector Space has the Same Dimension
Classification Theorem and Rank-Nullity
13.4
Matrices and Change of Basis
Matrices of Linear Transformations
Change of Basis
Elementary Basis Operations
14
Finitely Generated Modules over PIDs
14.1
Finitely Presented Modules
Module Presentations
When Modules are Finitely Presented: Noetherian Rings
14.2
Smith Normal Form
Stating and Finding the Smith Normal Form of a Matrix
Proof and Uniqueness of Smith Normal Form
14.3
Classifications
Invariant Factors
Elementary Divisors
15
Canonical Forms
15.1
Rational Canonical Form
Making Transformations into Modules
Rational Heads Prevail
15.2
The Cayley-Hamilton Theorem
15.3
Jordan Canonical Form
IV
Field Theory
16
Field Extensions
16.1
Irredicuble Polynomials
Basic Irreducibility Tests
Eisenstein’s Criterion
16.2
Field Extension Basics
Welcome to Field Extensions
Degree of a Field Extension
Simple and Generated Extensions
Uniqueness of Simple Extensions
16.3
Algebraic Extensions
Algebraic Elements
Algebraic Field Extensions
16.4
Algebraic Closures
Clooooosing Time
16.5
Splitting Fields
Split Up and Look For Clues
16.6
Separability
Ring Characteristic
Separable Polynomials
Separable Field Extensions
17
Galois Theory
17.1
Galois Extensions
Group Actions and Automorphism Groups
Finite Extensions and Galois Groups
17.2
The Fundamental Theorem of Galois Theory
The Fundamental Theorem: Statement and Uses
The Fundamental Theorem: Proof and Artin’s Theorem
Backmatter
A
Foundational Knowledge
A.1
Sets, Functions, Constructions
Sets
Functions
Set Constructions
Equivalence Relations and Modular Arithmetic
A.2
Numbers, Counting, Cardinality
Number Theory
Counting (*)
Cardinality
A.3
Posets, Lattices, Chains (*)
Orderings and Ordered Sets
B
Computational Tools: Sage
B.1
Sage and Groups
Sage and Groups
Sage and Finitely Presented Groups
B.2
GAP
What is GAP?
Commands in GAP
B.3
Macaulay2
What is Macaulay2?
Commands in Macaulay2
C
Qualifying Exams
C.1
Syllabus and Problems
C.2
June 2023
Section I: Group Theory
Section II: Rings, Modules, and Linear Algebra
Section III: Fields and Galois Theory
C.3
January 2023
Section I: Group Theory
Section II: Rings, Modules, and Linear Algebra
Section III: Fields and Galois Theory
C.4
May 2022
Section I: Group Theory
Section II: Rings, Modules, and Linear Algebra
Section III: Fields and Galois Theory
C.5
January 2022
Section I: Group Theory
Section II: Rings, Modules, and Linear Algebra
Section III: Fields and Galois Theory
C.6
May 2021
Section I: Group Theory
Section II: Rings, Modules, and Linear Algebra
Section III: Fields and Galois Theory
C.7
January 2021
Section I: Group Theory
Section II: Rings, Modules, and Linear Algebra
Section III: Fields and Galois Theory
C.8
June 2020
Section I: Group Theory
Section II: Rings, Modules, and Linear Algebra
Section III: Fields and Galois Theory
C.9
January 2020
Section I: Group Theory
Section II: Rings, Modules, and Linear Algebra
Section III: Fields and Galois Theory
C.10
May 2019
Section I: Group Theory
Section II: Rings, Modules, and Linear Algebra
Section III: Fields and Galois Theory
C.11
January 2019
Section I: Group Theory
Section II: Rings, Modules, and Linear Algebra
Section III: Fields and Galois Theory
C.12
May 2018
Section I: Group Theory
Section II: Rings, Modules, and Linear Algebra
Section III: Fields and Galois Theory
C.13
May 2017
Section I: Group Theory
Section II: Rings, Modules, and Linear Algebra
Section III: Fields and Galois Theory
C.14
January 2017
Section I: Group Theory
Section
Section
C.15
May 2016
Section I: Group Theory
Section
Section
C.16
January 2016
Section I: Group Theory
Section
Section
C.17
January 2015
Section I: Group Theory
Section
Section
C.18
June 2014
Section I: Group Theory
Section
Section III: Linear Algebra and Modules
C.19
January 2014
Section I: Group Theory
Section II: Field Theory
Section III: Rings, Modules, and Linear Algebra
C.20
June 2012
Section I: Group Theory
Section
Section
C.21
January 2012
Section I: Groups
Section II: Rings and Fields
Section III: Linear Algebra and Modules
C.22
June 2011
Section I: Groups
Section II: Rings and Fields
Section III: Linear Algebra and Modules
C.23
June 2010
Section I: Groups
Section II: Rings and Fields
Section III: Linear Algebra and Modules
Index
Colophon
🔗
Chapter
14
Finitely Generated Modules over PIDs
14.1
Finitely Presented Modules
14.2
Smith Normal Form
14.3
Classifications
