Gilbert L., Gilbert J. Elements of Modern Algebra

7th edition. — Brooks Cole, 2008. — 528 p."Elements of Modern Algebra" 7e, with its user-friendly format, provides you with the tools you need to get succeed in abstract algebra and develop mathematical maturity as a bridge to higher-level mathematics courses. Strategy boxes give you guidance and explanations about techniques and enable you to become more proficient at constructing proofs. A summary of key words and phrases at the end of each chapter help you master the material. A reference section, symbolic marginal notes, an appendix, and numerous examples help you develop your problem solving skills."Elements of Modern Algebra" is intended for an introductory course in abstract algebra taken by Math and Math for Secondary Education majors. Helping to make the study of abstract algebra more accessible, this text gradually introduces and develops concepts through helpful features that provide guidance on the techniques of proof construction and logic analysis. The text develops mathematical maturity for students by presenting the material in a theorem-proof format, with definitions and major results easily located through a user-friendly format. The treatment is rigorous and self-contained, in keeping with the objectives of training the student in the techniques of algebra and of providing a bridge to higher-level mathematical courses. The text has a flexible organization, with section dependencies clearly mapped out and optional topics that instructors can cover or skip based on their course needs. Additionally, problem sets are carefully arranged in order of difficulty to cater assignments to varying student ability levels.Fundamentals.
Sets.
Mappings.
Properties of Composite Mappings (Optional).
Binary Operations.
Permutations and Inverses.
Matrices.
Relations.
Key Words and Phrases
A Pioneer in Mathematics: Arthur Cayley
The Integers.
Postulates for the Integers (Optional).
Mathematical Induction.
Divisibility.
Prime Factors and Greatest Common Divisor.
Congruence of Integers.
Congruence Classes.
Introduction to Coding Theory (Optional).
Introduction to Cryptography (Optional).
Key Words and Phrases
A Pioneer in Mathematics: Blaise Pascal
Groups.
Definition of a Group.
Properties of Group Elements.
Subgroups. Cyclic Groups.
Isomorphisms.
Homomorphisms.
Key Words and Phrases
A Pioneer in Mathematics: Niels Henrik Abel
More on Groups.
Finite Permutation Groups.
Cayley's Theorem.
Permutation Groups in Science and Art (Optional).
Cosets of a Subgroup.
Normal Subgroups.
Quotient Groups.
Direct Sums (Optional).
Some Results on Finite Abelian Groups (Optional).
Key Words and Phrases
A Pioneer in Mathematics: Augustin Louis Cauchy
Rings, Integral Domains, and Fields.
Definition of a Ring.
Integral Domains and Fields.
The Field of Quotients of an Integral Domain.
Ordered Integral Domains.
Key Words and Phrases
A Pioneer in Mathematics: Richard Dedekind
More on Rings.
Ideals and Quotient Rings.
Ring Homomorphisms.
The Characteristic of a Ring.
Maximal Ideals (Optional).
Key Words and Phrases
A Pioneer in Mathematics: Amalie Emmy Noether
Real and Complex Numbers.
The Field of Real Numbers.
Complex Numbers and Quaternions.
De Moivre's Theorem and Roots of Complex Numbers.
Key Words and Phrases
A Pioneer in Mathematics: William Rowan Hamilton
Polynomials.
Polynomials over a Ring.
Divisibility and Greatest Common Divisor.
Factorization in F[x].
Zeros of a Polynomial.
Solution of Cubic and Quartic Equations by Formulas (Optional).
Algebraic Extensions of a Field.
Key Words and Phrases
A Pioneer in Mathematics: Carl Friedrich GaussAppendix: The Basics of Logic
Answers to True/False and Selected
Exercises

Index