抽象代数：探究型アプローチ（第２版） Abstract Algebra : An Inquiry-Based Approach (Textbooks in Mathematics) （2ND）

Hodge, Jonathan K./ Schlicker, Steven/ Sundstrom, Ted

Chapman & Hall/CRC（2023/12発売）

• 製本 Hardcover:ハードカバー版／ページ数 525 p.
• 言語 ENG
• 商品コード 9780367555016
• DDC分類 512.02

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