A First Course in Abstract Algebra: With Applications
Autor Joseph J. Rotmanen Limba Engleză Paperback – 30 sep 2005
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Specificații
ISBN-13: 9780131862678
ISBN-10: 0131862677
Pagini: 581
Dimensiuni: 181 x 243 x 28 mm
Greutate: 1.11 kg
Ediția:3Nouă
Editura: Prentice Hall
Locul publicării:Upper Saddle River, United States
ISBN-10: 0131862677
Pagini: 581
Dimensiuni: 181 x 243 x 28 mm
Greutate: 1.11 kg
Ediția:3Nouă
Editura: Prentice Hall
Locul publicării:Upper Saddle River, United States
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Cuprins
Chapter 1: Number Theory Induction
Binomial Coefficients
Greatest Common Divisors
The Fundamental Theorem of Arithmetic
Congruences
Dates and Days
Chapter 2: Groups I Some Set Theory
Permutations
Groups
Subgroups and Lagrange's Theorem
Homomorphisms
Quotient Groups
Group Actions
Counting with Groups
Chapter 3: Commutative Rings I First Properties
Fields
Polynomials
Homomorphisms
Greatest Common Divisors
Unique Factorization
Irreducibility
Quotient Rings and Finite Fields
Officers, Magic, Fertilizer, and Horizons
Chapter 4: Linear Algebra Vector Spaces
Euclidean Constructions
Linear Transformations
Determinants
Codes
Canonical Forms
Chapter 5: Fields Classical Formulas
Insolvability of the General Quintic
Epilog
Chapter 6: Groups II Finite Abelian Groups
The Sylow Theorems
Ornamental Symmetry
Chapter 7: Commutative Rings III Prime Ideals and Maximal Ideals
Unique Factorization
Noetherian Rings
Varieties
Grobner Bases
Hints for Selected Exercises
Bibliography
Index
Binomial Coefficients
Greatest Common Divisors
The Fundamental Theorem of Arithmetic
Congruences
Dates and Days
Chapter 2: Groups I Some Set Theory
Permutations
Groups
Subgroups and Lagrange's Theorem
Homomorphisms
Quotient Groups
Group Actions
Counting with Groups
Chapter 3: Commutative Rings I First Properties
Fields
Polynomials
Homomorphisms
Greatest Common Divisors
Unique Factorization
Irreducibility
Quotient Rings and Finite Fields
Officers, Magic, Fertilizer, and Horizons
Chapter 4: Linear Algebra Vector Spaces
Euclidean Constructions
Linear Transformations
Determinants
Codes
Canonical Forms
Chapter 5: Fields Classical Formulas
Insolvability of the General Quintic
Epilog
Chapter 6: Groups II Finite Abelian Groups
The Sylow Theorems
Ornamental Symmetry
Chapter 7: Commutative Rings III Prime Ideals and Maximal Ideals
Unique Factorization
Noetherian Rings
Varieties
Grobner Bases
Hints for Selected Exercises
Bibliography
Index
Caracteristici
• Comprehensive coverage of abstract algebra – Includes discussions of the fundamental theorem of Galois theory; Jordan-Holder theorem; unitriangular groups; solvable groups; construction of free groups; von Dyck's theorem, and presentations of groups by generators and relations.
• Significant applications for both group and commutative ring theories, especially with Gr ö bner bases – Helps students see the immediate value of abstract algebra.
• Flexible presentation – May be used to present both ring and group theory in one semester, or for two-semester course in abstract algebra.
• Number theory – Presents concepts such as induction, factorization into primes, binomial coefficients and DeMoivre's Theorem, so students can learn to write proofs in a familiar context.
• Section on Euclidean rings – Demonstrates that the quotient and remainder from the division algorithm in the Gaussian integers may not be unique. Also, Fermat's Two-Squares theorem is proved.
• Sylow theorems – Discusses the existence of Sylow subgroups as well as conjugacy and the congruence condition on their number.
• Fundamental theorem of finite abelian groups – Covers the basis theorem as well as the uniqueness to isomorphism
• Extensive references and consistent numbering system for lemmas, theorems, propositions, corollaries, and examples – Clearly organized notations, hints, and appendices simplify student reference.
• Significant applications for both group and commutative ring theories, especially with Gr ö bner bases – Helps students see the immediate value of abstract algebra.
• Flexible presentation – May be used to present both ring and group theory in one semester, or for two-semester course in abstract algebra.
• Number theory – Presents concepts such as induction, factorization into primes, binomial coefficients and DeMoivre's Theorem, so students can learn to write proofs in a familiar context.
• Section on Euclidean rings – Demonstrates that the quotient and remainder from the division algorithm in the Gaussian integers may not be unique. Also, Fermat's Two-Squares theorem is proved.
• Sylow theorems – Discusses the existence of Sylow subgroups as well as conjugacy and the congruence condition on their number.
• Fundamental theorem of finite abelian groups – Covers the basis theorem as well as the uniqueness to isomorphism
• Extensive references and consistent numbering system for lemmas, theorems, propositions, corollaries, and examples – Clearly organized notations, hints, and appendices simplify student reference.
Caracteristici noi
• Rewritten for smoother exposition – Makes challenging material more accessible to students.
• Updated exercises – Features challenging new problems, with redesigned page and back references for easier access.
• Extensively revised Ch. 2 (groups) and Ch. 3 (commutative rings ) – Makes chapters independent of one another, giving instructors increased flexibility in course design.
• New coverage of codes – Includes 28-page introduction to codes, including a proof that Reed-Solomon codes can be decoded.
• New section on canonical forms (Rational, Jordan, Smith) for matrices – Focuses on the definition and basic properties of exponentiation of complex matrices, and why such forms are valuable.
• New classification of frieze groups – Discusses why viewing the plane as complex numbers allows one to describe all isometries with very simple formulas.
• Expanded discussion of orthogonal Latin squares – Includes coverage of magic squares.
• Special Notation section – References common symbols and the page on which they are introduced.
• Updated exercises – Features challenging new problems, with redesigned page and back references for easier access.
• Extensively revised Ch. 2 (groups) and Ch. 3 (commutative rings ) – Makes chapters independent of one another, giving instructors increased flexibility in course design.
• New coverage of codes – Includes 28-page introduction to codes, including a proof that Reed-Solomon codes can be decoded.
• New section on canonical forms (Rational, Jordan, Smith) for matrices – Focuses on the definition and basic properties of exponentiation of complex matrices, and why such forms are valuable.
• New classification of frieze groups – Discusses why viewing the plane as complex numbers allows one to describe all isometries with very simple formulas.
• Expanded discussion of orthogonal Latin squares – Includes coverage of magic squares.
• Special Notation section – References common symbols and the page on which they are introduced.
Descriere
This text introduces students to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each.