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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

*S _{n}*: Our Main Example

Subgroups of *S _{n}*

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 *S _{n}*

The Alternating Group

Exercises

*S _{n}* and

*A*

_{n}

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[x_{1}, . . . ,x_{n}], 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