Algebra (Classic Version)

About our author 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.

• 1. Matrices 1.1 The Basic Operations1.2 Row Reduction1.3 The Matrix Transpose1.4 Determinants1.5 Permutations1.6 Other Formulas for the Determinant1.7 Exercises2. Groups 2.1 Laws of Composition2.2 Groups and Subgroups2.3 Subgroups of the Additive Group of Integers2.4 Cyclic Groups2.5 Homomorphisms2.6 Isomorphisms2.7 Equivalence Relations and Partitions2.8 Cosets2.9 Modular Arithmetic2.10 The Correspondence Theorem2.11 Product Groups2.12 Quotient Groups2.13 Exercises3. Vector Spaces 3.1 Subspaces of Rn3.2 Fields3.3 Vector Spaces3.4 Bases and Dimension3.5 Computing with Bases3.6 Direct Sums3.7 Infinite-Dimensional Spaces3.8 Exercises4. Linear Operators 4.1 The Dimension Formula4.2 The Matrix of a Linear Transformation4.3 Linear Operators4.4 Eigenvectors4.5 The Characteristic Polynomial4.6 Triangular and Diagonal Forms4.7 Jordan Form4.8 Exercises5. Applications of Linear Operators 5.1 Orthogonal Matrices and Rotations5.2 Using Continuity5.3 Systems of Differential Equations5.4 The Matrix Exponential5.5 Exercises6. Symmetry 6.1 Symmetry of Plane Figures6.2 Isometries6.3 Isometries of the Plane6.4 Finite Groups of Orthogonal Operators on the Plane6.5 Discrete Groups of Isometries6.6 Plane Crystallographic Groups6.7 Abstract Symmetry: Group Operations6.8 The Operation on Cosets6.9 The Counting Formula6.10 Operations on Subsets6.11 Permutation Representation6.12 Finite Subgroups of the Rotation Group6.13 Exercises7. More Group Theory 7.1 Cayley's Theorem7.2 The Class Equation7.3 r-groups7.4 The Class Equation of the Icosahedral Group7.5 Conjugation in the Symmetric Group7.6 Normalizers7.7 The Sylow Theorems7.8 Groups of Order 127.9 The Free Group7.10 Generators and Relations7.11 The Todd-Coxeter Algorithm7.12 Exercises8. Bilinear Forms 8.1 Bilinear Forms8.2 Symmetric Forms8.3 Hermitian Forms8.4 Orthogonality8.5 Euclidean spaces and Hermitian spaces8.6 The Spectral Theorem8.7 Conics and Quadrics8.8 Skew-Symmetric Forms8.9 Summary8.10 Exercises9. Linear Groups 9.1 The Classical Groups9.2 Interlude: Spheres9.3 The Special Unitary Group SU29.4 The Rotation Group SO39.5 One-Parameter Groups9.6 The Lie Algebra9.7 Translation in a Group9.8 Normal Subgroups of SL29.9 Exercises10. Group Representations 10.1 Definitions10.2 Irreducible Representations10.3 Unitary Representations10.4 Characters10.5 One-Dimensional Characters10.6 The Regular Representations10.7 Schur's Lemma10.8 Proof of the Orthogonality Relations10.9 Representationsof SU210.10 Exercises11. Rings 11.1 Definition of a Ring11.2 Polynomial Rings11.3 Homomorphisms and Ideals11.4 Quotient Rings11.5 Adjoining Elements11.6 Product Rings11.7 Fraction Fields11.8 Maximal Ideals11.9 Algebraic Geometry11.10 Exercises12. Factoring 12.1 Factoring Integers12.2 Unique Factorization Domains12.3 Gauss's Lemma12.4 Factoring Integer Polynomial12.5 Gauss Primes12.6 Exercises13. Quadratic Number Fields 13.1 Algebraic Integers13.2 Factoring Algebraic Integers13.3 Ideals in Z √(-5)13.4 Ideal Multiplication13.5 Factoring Ideals13.6 Prime Ideals and Prime Integers13.7 Ideal Classes13.8 Computing the Class Group13.9 Real Quadratic Fields13.10 About Lattices13.11 Exercises14. Linear Algebra in a Ring 14.1 Modules14.2 Free Modules14.3 Identities14.4 Diagonalizing Integer Matrices14.5 Generators and Relations14.6 Noetherian Rings14.7 Structure to Abelian Groups14.8 Application to Linear Operators14.9 Polynomial Rings in Several Variables14.10 Exercises15. Fields 15.1 Examples of Fields15.2 Algebraic and Transcendental Elements15.3 The Degree of a Field Extension15.4 Finding the Irreducible Polynomial15.5 Ruler and Compass Constructions15.6 Adjoining Roots15.7 Finite Fields15.8 Primitive Elements15.9 Function Fields15.10 The Fundamental Theorem of Algebra15.11 Exercises16. Galois Theory 16.1 Symmetric Functions16.2 The Discriminant16.3 Splitting Fields16.4 Isomorphisms of Field Extensions16.5 Fixed Fields16.6 Galois Extensions16.7 The Main Theorem16.8 Cubic Equations16.9 Quartic Equations16.10 Roots of Unity16.11 Kummer Extensions16.12 Quintic Equations16.13 ExercisesAppendix A. Background Material A.1 About ProofsA.2 The IntegersA.3 Zorn's LemmaA.4 The Implicit Function TheoremA.5 Exercises