|First Course in Abstract Algebra, A||
First Course in Abstract Algebra, A
|155.40||approx. 7-9 days|
This text introduces students to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each.
• Comprehensive coverage of abstract algebra - Includes discussions of the fundamental theorem of Galois theory; Jordan-Holder theorem; unitriangular groups; solvable groups; construction of free groups; von Dyck's theorem, and presentations of groups by generators and relations.
• Significant applications for both group and commutative ring theories, especially with Gr ö bner bases - Helps students see the immediate value of abstract algebra.
• Flexible presentation - May be used to present both ring and group theory in one semester, or for two-semester course in abstract algebra.
• Number theory - Presents concepts such as induction, factorization into primes, binomial coefficients and DeMoivre's Theorem, so students can learn to write proofs in a familiar context.
• Section on Euclidean rings - Demonstrates that the quotient and remainder from the division algorithm in the Gaussian integers may not be unique. Also, Fermat's Two-Squares theorem is proved.
• Sylow theorems - Discusses the existence of Sylow subgroups as well as conjugacy and the congruence condition on their number.
• Fundamental theorem of finite abelian groups - Covers the basis theorem as well as the uniqueness to isomorphism
• Extensive references and consistent numbering system for lemmas, theorems, propositions, corollaries, and examples - Clearly organized notations, hints, and appendices simplify student reference.
• Rewritten for smoother exposition - Makes challenging material more accessible to students.
• Updated exercises - Features challenging new problems, with redesigned page and back references for easier access.
• Extensively revised Ch. 2 (groups) and Ch. 3 (commutative rings ) - Makes chapters independent of one another, giving instructors increased flexibility in course design.
• New coverage of codes - Includes 28-page introduction to codes, including a proof that Reed-Solomon codes can be decoded.
• New section on canonical forms (Rational, Jordan, Smith) for matrices - Focuses on the definition and basic properties of exponentiation of complex matrices, and why such forms are valuable.
• New classification of frieze groups - Discusses why viewing the plane as complex numbers allows one to describe all isometries with very simple formulas.
• Expanded discussion of orthogonal Latin squares - Includes coverage of magic squares.
• Special Notation section - References common symbols and the page on which they are introduced.
Chapter 1: Number Theory
Greatest Common Divisors
The Fundamental Theorem of Arithmetic
Dates and Days
Chapter 2: Groups I
Some Set Theory
Subgroups and Lagrange's Theorem
Counting with Groups
Chapter 3: Commutative Rings I
Greatest Common Divisors
Quotient Rings and Finite Fields
Officers, Magic, Fertilizer, and Horizons
Chapter 4: Linear Algebra
Chapter 5: Fields
Insolvability of the General Quintic
Chapter 6: Groups II
Finite Abelian Groups
The Sylow Theorems
Chapter 7: Commutative Rings III
Prime Ideals and Maximal Ideals
Hints for Selected Exercises