عنوان

Introduction to Algebra

9400981791

9400981813

9789400981799

9789400981812

b593955

Introduction to Algebra

[Book]

by R. Kochendörffer.

Dordrecht

Springer Netherlands

1981

1. Basic concepts.- 1.1. Sets.- 1.2. Relations.- 1.3. Mappings.- 1.4. Operations.- 1.5. Algebraic systems.- Exercises.- 2. The integers.- 2.1. The natural numbers and the integers.- 2.2. Divisibility. Prime numbers.- 2.3. The greatest common divisor.- 2.4. Prime factorization.- 2.5. Congruences. Residue classes.- 2.6. The residue class ring.- 2.7. Simultaneous congruences. Euler's function.- Exercises.- 3. Groups.- 3.1. Semigroups.- 3.2. Groups.- 3.3. Isomorphisms. Automorphisms.- 3.4. Embedding of abelian semigroups in groups.- 3.5. Subgroups.- 3.6. Cyclic groups.- 3.7. Homomorphisms.- 3.8. Subnormal series.- 3.9. Direct products.- 3.10. Permutation groups.- 3.11. Sylow subgroups and p-groups.- 3.12. Endomorphisms and operators.- 3.13. Vector spaces. Modules.- Exercises.- 4. Rings. Integral domains.- 4.1. Definitions and examples.- 4.2. Homomorphisms.- 4.3. Commutative rings. Integral domains.- 4.4. Principal ideal rings.- 4.5. Euclidean rings.- 4.6. Fields of quotients.- 4.7. Prime fields. Characteristic.- Exercises.- 5. Polynomials.- 5.1. Polynomials in one indeterminate.- 5.2. Polynomials over fields.- 5.3. Polynomials over integral domains..- 5.4. Roots. The derivative.- 5.5. Polynomials in several indeterminates.- 5.6. Symmetric polynomials.- 5.7. The resultant and the discriminant.- Exercises.- 6. Fields.- 6.1. Adjunction.- 6.2. Algebraic extension fields.- 6.3. Construction of extension fields.- 6.4. Normal extensions.- 6.5. Separable and inseparable extensions.- 6.6. Galois theory.- 6.7. Cyclotomic fields.- 6.8. Galois fields.- Exercises.- 7. Galois theory of equations.- 7.1. The Galois group of a polynomial.- 7.2. Solubility of equations by radicals.- 7.3. Quadratic, cubic, and quartic equations.- 7.4. Constructions by ruler and compass.- Exercises.- 8. Order and valuations.- 8.1. Ordered fields.- 8.2. Formally real fields.- 8.3. Valuations.- Exercises.- 9. Modules.- 9.1. Elementary divisors.- 9.2. Modules over principal ideal rings.- 9.3. Endomorphisms of vector spaces.- 9.4. Finiteness conditions.- 9.5. Algebraic integers.- Exercises.- 10. Algebras.- 10.1. Basic definitions.- 10.2. The radical.- 10.3. Semi-simple rings.- 10.4. Simple rings.- 10.5. Division algebras over the field of the real numbers.- 10.6. Representation modules.- 10.7. Representations of semi-simple algebras.- Exercises.- 11. Lattices.- 11.1. Lattices and partially ordered sets.- 11.2. Modular lattices.- 11.3. Distributive lattices.- Exercises.

Mathematics.

by R. Kochendörffer.

R Kochendörffer

[Book]

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