Introduction to Algebra

ISBN-13: 9789400981812 / Angielski / Miękka / 2011 / 414 str.

R. Kochendorffer

Opis

This book is intended as a textbook for an undergraduate course on algebra. In most universities a detailed study *of abstract algebraic systems commences in the second year. By this time the student has gained some experience in mathematical reasoning so that a too elementary book would rob him of the joy and the stimulus of using his ability. I tried to make allowance for this when I chose t4e level of presentation. On the other hand, I hope that I also avoided discouraging the reader by demands which are beyond his strength. So, the first chapters will certainly not require more mathematical maturity than can reasonably be expected after the first year at the university. Apart from one exception the formal prerequisites do not exceed the syllabus of an average high school. As to the exception, I assume that the reader is familiar with the rudiments of linear algebra, i. e. addition and multiplication of matrices and the main properties of determinants. In view of the readers for whom the book is designed I felt entitled to this assumption. In the first chapters, matrices will almost exclusively occur in examples and exercises providing non-trivial instances in the theory of groups and rings. In Chapters 9 and 10 only, vector spaces and their properties will form a relevant part of the text. A reader who is not familiar with these concepts will have no difficulties in acquiring these prerequisites by any elementary textbook, e. g. [10].

Spis treści

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 greatést 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.