1st Edition

Algebra Groups, Rings, and Fields

By Louis Rowen Copyright 1994
    264 Pages
    by A K Peters/CRC Press

    264 Pages
    by A K Peters/CRC Press

    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