2nd Edition

# Abstract Algebra An Inquiry-Based Approach

**Also available as eBook on:**

**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.

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

### Biography

**Jonathan K. Hodge **is a Professor of Mathematics and Dean of the School of Natural Sciences at St. Edward's University. Prior to joining SEU, Dr. Hodge taught mathematics for 19 years at Grand Valley State University, where he also co-directed GVSU's Summer Mathematics Research Experience for Undergraduates (REU). Dr. Hodge earned his Ph.D. in mathematics from Western Michigan University in 2002. In addition to* Abstract Algebra: An Inquiry-Based Approach*, he is also a co-author of *The Mathematics of Voting and Elections: A Hands-On Approach*, published by the American Mathematical Society.

**Steven Schlicker** is a Professor of Mathematics at Grand Valley State University in Allendale, MI. He earned his bachelor’s degree in mathematics from Michigan State University and a Ph.D. in mathematics (in group cohomology and algebraic K-theory) from Northwestern University. In addition to being a coauthor of Abstract Algebra: An Inquiry-Based Approach, he is a contributing author to Active Calculus, the primary author of Active Calculus Multivariable, and coauthor of the texts Discovering Wavelets and Linear Algebra and Applications: An Inquiry-based Approach, and of a trigonometry book with Ted Sundstrom.

**Ted Sundstrom** is Professor Emeritus of Mathematics at Grand Valley State University having retired in 2017 after 44 years of service. He received a bachelor’s degree in mathematics from Western Michigan University in 1968 and a Ph.D. in mathematics (ring theory) from the University of Massachusetts in 1973. In 2005, he received the award for Distinguished Teaching of College or University Mathematics by the Michigan Section of the MAA. Besides being a coauthor of Abstract Algebra: An Inquiry-Based Approach, he is the author of Mathematical Reasoning: Writing and Proof, and coauthored a trigonometry book with Steven Schlicker. In 2017, Prof. Sundstrom was the inaugural recipient of the Daniel Solow Author’s Award from the Mathematical Association of America for the book Mathematical Reasoning: Writing and Proof.