Numbers, Polynomials, and Factoring The Natural Numbers The Integers Modular Arithmetic Polynomials with Rational CoefficientsFactorization of PolynomialsSection I in a NutshellRings, Domains, and Fields Rings Subrings and Unity Integral Domains and Fields Ideals Polynomials over a Field Section II in a NutshellRing Homomorphisms and Ideals Ring HomomorphismsThe Kernel Rings of Cosets The Isomorphism Theorem for Rings Maximal and Prime Ideals The Chinese Remainder Theorem Section III in a NutshellGroups Symmetries of Geometric Figures PermutationsAbstract Groups Subgroups Cyclic Groups Section Read more...
Abstract: Numbers, Polynomials, and Factoring The Natural Numbers The Integers Modular Arithmetic Polynomials with Rational CoefficientsFactorization of PolynomialsSection I in a NutshellRings, Domains, and Fields Rings Subrings and Unity Integral Domains and Fields Ideals Polynomials over a Field Section II in a NutshellRing Homomorphisms and Ideals Ring HomomorphismsThe Kernel Rings of Cosets The Isomorphism Theorem for Rings Maximal and Prime Ideals The Chinese Remainder Theorem Section III in a NutshellGroups Symmetries of Geometric Figures PermutationsAbstract Groups Subgroups Cyclic Groups Section
Content: Front Cover
Contents
Preface
Part I: Numbers, Polynomials, and Factoring
Chapter 1: The Natural Numbers
Chapter 2: The Integers
Chapter 3: Modular Arithmetic
Chapter 4: Polynomials with Rational Coefficients
Chapter 5: Factorization of Polynomials
Section I: in a Nutshell
Part II: Rings, Domains, and Fields
Chapter 6: Rings
Chapter 7: Subrings and Unity
Chapter 8: Integral Domains and Fields
Chapter 9: Ideals
Chapter 10: Polynomials over a Field
Section II: in a Nutshell
Part III: Ring Homomorphisms and Ideals
Chapter 11: Ring Homomorphisms
Chapter 12: The Kernel. Chapter 13: Rings of CosetsChapter 14: The Isomorphism Theorem for Rings
Chapter 15: Maximal and Prime Ideals
Chapter 16: The Chinese Remainder Theorem
Section III: in a Nutshell
Part IV: Groups
Chapter 17: Symmetries of Geometric Figures
Chapter 18: Permutations
Chapter 19: Abstract Groups
Chapter 20: Subgroups
Chapter 21: Cyclic Groups
Section IV: in a Nutshell
Part V: Group Homomorphisms
Chapter 22: Group Homomorphisms
Chapter 23: Structure and Representation
Chapter 24: Cosets and Lagrange's Theorem
Chapter 25: Groups of Cosets. Chapter 26: The Isomorphism Theorem for GroupsSection V: in a Nutshell
Part VI: Topics from Group Theory
Chapter 27: The Alternating Groups
Chapter 28: Sylow Theory: The Preliminaries
Chapter 29: Sylow Theory: The Theorems
Chapter 30: Solvable Groups
Section VI: in a Nutshell
Part VII: Unique Factorization
Chapter 31: Quadratic Extensions of the Integers
Chapter 32: Factorization
Chapter 33: Unique Factorization
Chapter 34: Polynomials with Integer Coefficients
Chapter 35: Euclidean Domains
Section VII: in a Nutshell
Part VIII: Constructibility Problems. Chapter 36: Constructions with Compass and StraightedgeChapter 37: Constructibility and Quadratic Field Extensions
Chapter 38: The Impossibility of Certain Constructions
Section VIII: in a Nutshell
Part IX: Vector Spaces and Field Extensions
Chapter 39: Vector Spaces I
Chapter 40: Vector Spaces II
Chapter 41: Field Extensions and Kronecker's Theorem
Chapter 42: Algebraic Field Extensions
Chapter 43: Finite Extensions and Constructibility Revisited
Section IX: in a Nutshell
Part X: Galois Theory
Chapter 44: The Splitting Field
Chapter 45: Finite Fields
Chapter 46: Galois Groups. Chapter 47: The Fundamental Theorem of Galois TheoryChapter 48: Solving Polynomials by Radicals
Section X: in a Nutshell
Hints and Solutions
Guide to Notation.
