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Full Description
Abstract Algebra: An Inquiry-Based Approach, Second Edition not only teaches abstract algebra, but also provides a deeper understanding of what mathematics is, how it is done, and how mathematicians think.
The second edition of this unique, flexible approach builds on the success of the first edition. The authors offer an emphasis on active learning, helping students learn algebra by gradually building both their intuition and their ability to write coherent proofs in context.
The goals for this text include:
Allowing the flexibility to begin the course with either groups or rings
Introducing the ideas behind definitions and theorems to help students develop intuition
Helping students understand how mathematics is done. Students will experiment through examples, make conjectures, and then refine or prove their conjectures
Assisting students in developing their abilities to effectively communicate mathematical ideas
Actively involving students in realizing each of these goals through in-class and out-of-class activities, common in-class intellectual experiences, and challenging problem sets
Changes in the Second Edition
Streamlining of introductory material with a quicker transition to the material on rings and groups
New investigations on extensions of fields and Galois theory
New exercises added and some sections reworked for clarity
More online Special Topics investigations and additional Appendices, including new appendices on other methods of proof and complex roots of unity
Encouraging students to do mathematics and be more than passive learners, this text shows students the way mathematics is developed is often different than how it is presented; definitions, theorems, and proofs do not simply appear fully formed; mathematical ideas are highly interconnected; and in abstract algebra, there is a considerable amount of intuition to be found.
Contents
I. Number Systems
1.The Integers
2. Equivalence Relations and [Equation]n
3. Algebra in Other Number Systems
II Rings
4. An Introduction to Rings
5. Integer Multiples and Exponents
6. Subrings, Extensions, and Direct Sums
7. Isomorphism and Invariants
III Polynomial Rings
8 Polynomial Rings
9 Divisibility in Polynomial Rings
10 Roots, Factors, and Irreducible Polynomials
11 Irreducible Polynomials
12 Quotients of Polynomial Rings
IV More Ring Theory
13 Ideals and Homomorphisms
14 Divisibility and Factorization in Integral Domains
15 From [Equation] to [Equation]
V Groups
16 Symmetry
17 An Introduction to Groups
18 Integer Powers of Elements in a Group
19 Subgroups
20 Subgroups of Cyclic Groups
21 The Dihedral Groups
22 The Symmetric Groups
23 Cosets and Lagrange's Theorem
24 Normal Subgroups and Quotient Groups
25 Products of Groups
26 Group Isomorphisms and Invariants
27 Homomorphisms and Isomorphism Theorems
28 The Fundamental Theorem of Finite Abelian Groups
29 The First Sylow Theorem
30 The Second and Third Sylow Theorems
VI Fields and Galois Theory
31 Finite Fields, the Group of Units in [Equation]n, and Splitting Fields
32 Extensions of Fields
33 Galois Theory
