Author | Kochendรถrffer, R. author |
---|---|
Title | Introduction to Algebra [electronic resource] / by R. Kochendรถrffer |
Imprint | Dordrecht : Springer Netherlands, 1972 |
Connect to | http://dx.doi.org/10.1007/978-94-009-8179-9 |
Descript | X, 414 p. online resource |
1. Basic concepts -- 1.1. Sets -- 1.2. Relations -- 1.3. Mappings -- 1.4. Operations -- 1.5. Algebraic systems -- 2. The integers -- 2.1. The natural numbers and the integers -- 2.2. Divisibility. Prime numbers -- 2.3. The greatรฉst common divisor -- 2.4. Prime factorization -- 2.5. Congruences. Residue classes -- 2.6. The residue class ring -- 2.7. Simultaneous congruences. Eulerโs function -- 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 -- 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 -- 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 -- 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 -- 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 -- 8. Order and valuations -- 8.1. Ordered fields -- 8.2. Formally real fields -- 8.3. Valuations -- 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 -- 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 -- 11. Lattices -- 11.1. Lattices and partially ordered sets -- 11.2. Modular lattices -- 11.3. Distributive lattices