اصول جبر انتزاعی: نسخه‌ای گسترش‌یافته

Title Contents Preface 1. Some fundamental concepts 1.1. Sets 1.2. Mappings 1.3. Products of mappings 1.4. Equivalence relations 1.5. Operations 2. Rings 2.1. Formal properties of the integers 2.2. Definition of a ring 2.3. Examples of rings 2.4. Some properties of addition 2.5. Some other properties of a ring 2.6. General sums and products 2.7. Homomorphisms and isomorphisms 3. Integral domains 3.1. Definition of an integral domain 3.2. Ordered integral domains 3.3. Well-ordering and mathematical induction 3.4. A characterization of the ring of integers 3.5. The Peano axioms (optional) 4. Some properties of the integers 4.1. Divisors and the division algorithm 4.2. Different bases (optional) 4.3. Greatest common divisor 4.4. The fundamental theorem 4.5. Some applications of the fundamental theorem 4.6. Pythagorean triples (optional) 4.7. The ring of integers modulo n 5. Fields and the rational numbers 5.1. Fields 5.2. The characteristic 5.3. Some familiar notation 5.4. The field of rational numbers 5.5. A few properties of the field of rational numbers 5.6. Subfields and extensions 5.7. Construction of the integers from the natural numbers (optional) 6. Real and complex numbers 6.1. The field of real numbers 6.2. Some properties of the field of real numbers 6.3. The field of complex numbers 6.4. The conjugate of a complex number 6.5. Geometric representation and trigonometric form 6.6. The nth roots of a complex number 7. Groups 7.1. Definition and simple properties 7.2. Groups of permutations 7.3. Homomorphisms and isomorphisms 7.4. Cyclic groups 7.5. Cosets and Lagrange's theorem 7.6. The symmetric group S_n 7.7. Normal subgroups and quotient groups 7.8. Homomorphisms and subgroups 8. Finite abelian groups 8.1. Direct sums of subgroups 8.2. Cyclic subgroups and bases 8.3. Finite abelian p-groups 8.4. The principal theorems for finite abelian groups 9. The Sylow theorems 9.1. Conjugate theorems and transforms 9.2. Conjugate subgroups 9.3. Double cosets 9.4. Proofs of the Sylow theorems 10. Polynomials 10.1. Polynomial rings 10.2. The substitution process 10.3. Divisors and the division algorithm 10.4. Greatest common divisor 10.5. Unique factorization in F[x] 10.6. Rational roots of a polynomial over the rational field 10.7. Prime polynomials over the rational field (optional) 10.8. Polynomials over the real or complex numbers 10.9. Partial fractions (optional) 11. Ideals and quotient rings 11.1. Ideals 11.2. Quotient rings 11.3. Quotient rings F[x] / (s(x)) 11.4. The fundamental theorem on ring homomorphisms 12. Vector spaces 12.1. Vectors in a plane 12.2. Definition and simple properties of a vector space 12.3. Linear dependence 12.4. Linear combinations and subspaces 12.5. Basis and dimension 12.6. Homomorphisms of vector spaces 12.7. Hom_F (V, W) as a vector space 12.8. Dual vector spaces 12.9. Quotient vector spaces and direct sums 12.10. Inner products in V_n (F) 13. Field extensions 13.1. The process of adjunction 13.2. The existence of certain extensions 13.3. Classifications of extensions 13.4. Simple extensions 13.5. Finite algebraic extensions 13.6. Equivalence of splitting fields of a polynomial 14. Systems of linear equations 14.1. Notation and simple results 14.2. Echelon systems 14.3. Matrices 14.4. Applications to systems of linear equations 14.5. Systems of linear homogeneous equations 15. Determinants 15.1. Preliminary remarks 15.2. General definition of determinant 15.3. Some fundamental properties 15.4. Expansion in terms of a row or column 15.5. The determinant rank of a matrix 15.6. Systems of linear equations 16. Linear transformations and matrices 16.1. Notation and preliminary remarks 16.2. Algebra of linear transformations 16.3. The finite dimensional case 16.4. Algebra of matrices 16.5. Linear transformations of V_n (F) 16.6. Adjoint and inverse of a matrix 16.7. Equivalence of matrices 16.8. The determinant of a product 16.9. Similarity of matrices 16.10. Invariant subspaces 16.11. Polynomials in a linear transformation 16.12. Characteristic vectors and characteristic roots 17. Some additional topics 17.1. Quaternions 17.2. Principal ideal domains 17.3. Modules 17.4. Modules over a principal ideal domain 17.5. Zorn's lemma 17.6. Representations of Boolean rings Bibliography Index