This text presents the concepts of higher algebra in a comprehensive and modern way for self-study and as a basis for a high-level undergraduate course. The author is one of the preeminent researchers in this field and brings the reader up to the recent frontiers of research including never-before-published material. From the table of contents: - Groups: Monoids and Groups - Cauchyís Theorem - Normal Subgroups - Classifying Groups - Finite Abelian Groups - Generators and Relations - When Is a Group a Group? (Cayley's Theorem) - Sylow Subgroups - Solvable Groups - Rings and Polynomials: An Introduction to Rings - The Structure Theory of Rings - The Field of Fractions - Polynomials and Euclidean Domains - Principal Ideal Domains - Famous Results from Number Theory - I Fields: Field Extensions - Finite Fields - The Galois Correspondence - Applications of the Galois Correspondence - Solving Equations by Radicals - Transcendental Numbers: e and p - Skew Field Theory - Each chapter includes a set of exercises
Preface
Table of Principal Notation
Prerequisites
Exercises
PART I—GROUPS
Monoids and Groups
Examples of Groups and Monoids
When Is a Monoid a Group?
Exercises
How to Divide: Lagrange’s Theorem, Cosets, and an Application to Number Theory
Cosets
Fermat’s Little Theorem
Exercises
Euler’s Number
Cauchy’s Theorem: How to Show a Number Is Greater Than 1
The Exponenet
Sn: Our Main Example
Subgroups of Sn
Cycles
The Product of Two Subgroups
The Classical Groups
Exercises
The Classical Groups
Introduction to the Classifications of Groups: Homomorphisms, Isomorphisms, and Invariants
Homomorphic Images
Exercises
Normal Subgroups—The Building Blocks of the Structure Theory
The Residue Group
Noether’s Isomorphism Theorems
Conjugates in Sn
The Alternating Group
Exercises
Sn and An
Classifying Groups—Cyclic Groups and Direct Products
Cyclic Groups
Generators of a Group
Direct Products
Internal Direct Products
Exercises
Finite Abelian Groups
Abelian p-Groups
Proof of the Fundamental Theorem for Finite Abelian Groups
The Classification of Finite Abelian Groups
Exercises
Finitely Generated Abelian Groups
Generators and Relations
Description of Groups of Low Order
Addendum: Erasing Relations
Exercises
Explicit Generation of Groups by Arbitrary Subsets
When Is a Group a Group? (Cayley’s Theorem)
Generalized Cayley’s Theorem
Group Representations
Exercises
Recounting: Conjugacy Classes and the Class Formula
The Center of a Group
Groups Acting on Sets: A Recapitulation
Exercises
Double Cosets
Group Actions on Sets
Sylow Subgroups: A New Invariant
Groups of Order Less Than 60
Simple Groups
Exercises
Classification of Groups of Various Orders
Solvable Groups: What Could Be Simpler?
Commutators
Solvable Groups
Addendum: Automorphisms of Groups
Exercises
Nilpotenet Groups
The Special Linear Group SL(n, F)
The Projective Special Linear Group PSL (n, F)
Exercises for the Addendum
Semidirect Products, also cf. Example 8.8
The Wreath Product
Review Exercises for Part I
PART II—RINGS AND POLYNOMIALS
An Introduction to Rings
Domains and Skew Fields
Left Ideals
Exercises
Rings of Matrices
Direct Products of Rings
The Structure Theory of Rings
Ideals
Noether’s Isomorphism Theorems
Exercises
The Regular Representation
General Structure Theory
The Field of Fractions—A Study in Generalization
Intermediate Rings
Exercises
Subrings of Q
Polynomials and Euclidean Domains
The Ring of Polynomials
Euclidean Domains
Unique Factorization
Exercises
Formal Power Series
The Partition Number
Unique Factorization Domains
Principal Ideal Domains: Induction without Numbers
Prime Ideals
Noetherian Rings
Exercises
Counterexamples
Consequences of Zorn’s Lemma
UFD’s
Noetherian Rings
Roots of Polynomials
Finite Subgroups of Fields
Primitive Roots of 1
Exercises
The Structure of Euler (n)
(Optional) Applications: Famous Results from Number Theory
A Theorem of Fermat
Addendum: Fermat’s Last Theorem
Exercises
Irreducible Polynomials
Polynomials over UFD’s
Einstein’s Criterion
Exercises
Nagata’s Theorem and its Applications
The Ring Z[x1, . . . ,xn], and the Generic Method
Review Exercises for Part II
PART III—FIELDS
Historical Background
Field Extensions: Creating Roots of Polynomials
Algebraic Elements
Finite Field Extensions
Exercises
Countability and Transcendental Numbers
Algebraic Extensions
The Problems of Antiquity
Construction by Straight Edge and Compass
Algebraic Description of Constructability
Solution of the Problems of Antiquity
Exercises
Constructability
Constructing a Regular n-gon
Adjoining Roots to Polynomials: Splitting Fields
Splitting Fields
Separable Polynomials and Separable Extensions
The Characteristic of a Field
Exercises
The Roots of a Polynomial in Terms of the Coefficients
Separability and the Characteristic
Calculus through the Looking Glass
Finite Fields
Reduction Modulo p
Exercises
The Galois Correspondence
The Galois Group of Automorphisms of a Field Extension
The Galois Group and Intermediate Fields
Exercises
The Galois Group of the Compositum
More on Artin’s Lemma
Applications of the Galois Correspondence
Finite Separable Field Extensions and the Normal Closure
The Galois Group of a Polynomial
Constructible n-gons
Finite Fields
The Fundamental Theorem of Algebra
Exercises
The Normal Closure
Separability Degree
Finite Fields
The Algebraic Closure
Solving Equations by Radicals
Root Extensions
Solvable Galois Groups
Computing the Galois Group
Exercises
Prescribed Galois Groups
Root Towers
Finding the Number of Real Roots via the Discriminant
Galois Groups and Solvability
The Norm and Trace
Review Exercises for Part III
Appendix A. Transcendental Numbers: e and π
Transcendence of e
Transcendence of π
Skew Field Theory
The Quaternion Algebra
Proof of Lagrange’s Four Square Theorem
Polynomials over Skew Fields
Structure Theorems for Skew Fields
Exercises
Proof of Lagrange’s Four Square Theorem
Frobenius’ Theorem
Index
Biography
Rowen \, Louis