Algebra (Classic Version)
Description
Appropriate for one- or two-semester algebra courses
This title is part of the Pearson Modern Classics series. Pearson Modern Classics are acclaimed titles at a value price. Please visit www.pearsonhighered.com/math-classics-series for a complete list of titles.
Algebra , 2nd Edition , by Michael Artin, provides comprehensive coverage at the level of an honors-undergraduate or introductory-graduate course. The second edition of this classic text incorporates twenty years of feedback plus the author's own teaching experience. This book discusses concrete topics of algebra in greater detail than others, preparing readers for the more abstract concepts; linear algebra is tightly integrated throughout.
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Michael Artin received his A.B. from Princeton University in 1955 and his M.A. and Ph.D. from Harvard University in 1956 and 1960, respectively. He continued at Harvard as Benjamin Peirce Lecturer, 1960 - 63. He joined the MIT mathematics faculty in 1963, and was appointed Norbert Wiener Professor from 1988 - 93. Outside MIT, Artin served as President of the American Mathematical Society from 1990-92. He has received honorary doctorate degrees from the University of Antwerp and University of Hamburg.
Professor Artin is an algebraic geometer, concentrating on non-commutative algebra. He has received many awards throughout his distinguished career, including the Undergraduate Teaching Prize and the Educational and Graduate Advising Award. He received the Leroy P. Steele Prize for Lifetime Achievement from the AMS. In 2005 he was honored with the Harvard Graduate School of Arts & Sciences Centennial Medal, for being "an architect of the modern approach to algebraic geometry." Professor Artin is a Member of the National Academy of Sciences, Fellow of the American Academy of Arts & Sciences, Fellow of the American Association for the Advancement of Science, and Fellow of the Society of Industrial and Applied Mathematics. He is a Foreign Member of the Royal Holland Society of Sciences, and Honorary Member of the Moscow Mathematical Society.
Content
- 1.1 The Basic Operations
- 1.2 Row Reduction
- 1.3 The Matrix Transpose
- 1.4 Determinants
- 1.5 Permutations
- 1.6 Other Formulas for the Determinant
- 1.7 Exercises
- 2.1 Laws of Composition
- 2.2 Groups and Subgroups
- 2.3 Subgroups of the Additive Group of Integers
- 2.4 Cyclic Groups
- 2.5 Homomorphisms
- 2.6 Isomorphisms
- 2.7 Equivalence Relations and Partitions
- 2.8 Cosets
- 2.9 Modular Arithmetic
- 2.10 The Correspondence Theorem
- 2.11 Product Groups
- 2.12 Quotient Groups
- 2.13 Exercises
- 3.1 Subspaces of R n
- 3.2 Fields
- 3.3 Vector Spaces
- 3.4 Bases and Dimension
- 3.5 Computing with Bases
- 3.6 Direct Sums
- 3.7 Infinite-Dimensional Spaces
- 3.8 Exercises
- 4.1 The Dimension Formula
- 4.2 The Matrix of a Linear Transformation
- 4.3 Linear Operators
- 4.4 Eigenvectors
- 4.5 The Characteristic Polynomial
- 4.6 Triangular and Diagonal Forms
- 4.7 Jordan Form
- 4.8 Exercises
- 5.1 Orthogonal Matrices and Rotations
- 5.2 Using Continuity
- 5.3 Systems of Differential Equations
- 5.4 The Matrix Exponential
- 5.5 Exercises
- 6.1 Symmetry of Plane Figures
- 6.2 Isometries
- 6.3 Isometries of the Plane
- 6.4 Finite Groups of Orthogonal Operators on the Plane
- 6.5 Discrete Groups of Isometries
- 6.6 Plane Crystallographic Groups
- 6.7 Abstract Symmetry: Group Operations
- 6.8 The Operation on Cosets
- 6.9 The Counting Formula
- 6.10 Operations on Subsets
- 6.11 Permutation Representation
- 6.12 Finite Subgroups of the Rotation Group
- 6.13 Exercises
- 7.1 Cayley's Theorem
- 7.2 The Class Equation
- 7.3 r -groups
- 7.4 The Class Equation of the Icosahedral Group
- 7.5 Conjugation in the Symmetric Group
- 7.6 Normalizers
- 7.7 The Sylow Theorems
- 7.8 Groups of Order 12
- 7.9 The Free Group
- 7.10 Generators and Relations
- 7.11 The Todd-Coxeter Algorithm
- 7.12 Exercises
- 8.1 Bilinear Forms
- 8.2 Symmetric Forms
- 8.3 Hermitian Forms
- 8.4 Orthogonality
- 8.5 Euclidean spaces and Hermitian spaces
- 8.6 The Spectral Theorem
- 8.7 Conics and Quadrics
- 8.8 Skew-Symmetric Forms
- 8.9 Summary
- 8.10 Exercises
- 9.1 The Classical Groups
- 9.2 Interlude: Spheres
- 9.3 The Special Unitary Group SU 2
- 9.4 The Rotation Group SO 3
- 9.5 One-Parameter Groups
- 9.6 The Lie Algebra
- 9.7 Translation in a Group
- 9.8 Normal Subgroups of SL 2
- 9.9 Exercises
- 10.1 Definitions
- 10.2 Irreducible Representations
- 10.3 Unitary Representations
- 10.4 Characters
- 10.5 One-Dimensional Characters
- 10.6 The Regular Representations
- 10.7 Schur's Lemma
- 10.8 Proof of the Orthogonality Relations
- 10.9 Representationsof SU 2
- 10.10 Exercises
- 11.1 Definition of a Ring
- 11.2 Polynomial Rings
- 11.3 Homomorphisms and Ideals
- 11.4 Quotient Rings
- 11.5 Adjoining Elements
- 11.6 Product Rings
- 11.7 Fraction Fields
- 11.8 Maximal Ideals
- 11.9 Algebraic Geometry
- 11.10 Exercises
- 12.1 Factoring Integers
- 12.2 Unique Factorization Domains
- 12.3 Gauss's Lemma
- 12.4 Factoring Integer Polynomial
- 12.5 Gauss Primes
- 12.6 Exercises
- 13.1 Algebraic Integers
- 13.2 Factoring Algebraic Integers
- 13.3 Ideals in Z (-5)
- 13.4 Ideal Multiplication
- 13.5 Factoring Ideals
- 13.6 Prime Ideals and Prime Integers
- 13.7 Ideal Classes
- 13.8 Computing the Class Group
- 13.9 Real Quadratic Fields
- 13.10 About Lattices
- 13.11 Exercises
- 14.1 Modules
- 14.2 Free Modules
- 14.3 Identities
- 14.4 Diagonalizing Integer Matrices
- 14.5 Generators and Relations
- 14.6 Noetherian Rings
- 14.7 Structure to Abelian Groups
- 14.8 Application to Linear Operators
- 14.9 Polynomial Rings in Several Variables
- 14.10 Exercises
- 15.1 Examples of Fields
- 15.2 Algebraic and Transcendental Elements
- 15.3 The Degree of a Field Extension
- 15.4 Finding the Irreducible Polynomial
- 15.5 Ruler and Compass Constructions
- 15.6 Adjoining Roots
- 15.7 Finite Fields
- 15.8 Primitive Elements
- 15.9 Function Fields
- 15.10 The Fundamental Theorem of Algebra
- 15.11 Exercises
- 16.1 Symmetric Functions
- 16.2 The Discriminant
- 16.3 Splitting Fields
- 16.4 Isomorphisms of Field Extensions
- 16.5 Fixed Fields
- 16.6 Galois Extensions
- 16.7 The Main Theorem
- 16.8 Cubic Equations
- 16.9 Quartic Equations
- 16.10 Roots of Unity
- 16.11 Kummer Extensions
- 16.12 Quintic Equations
- 16.13 Exercises
- A.1 About Proofs
- A.2 The Integers
- A.3 Zorn's Lemma
- A.4 The Implicit Function Theorem
- A.5 Exercises