Abstract Algebra

An Inquiry-Based Approach

Inbunden, Engelska, 2023

Av Jonathan K. Hodge, Steven Schlicker, Ted Sundstrom, USA) Hodge, Jonathan K. (Grand Valley State University, Allendale, Michigan, USA) Schlicker, Steven (Grand Valley State University, Allendale, Michigan, USA) Sundstrom, Ted (Grand Valley State University, Allendale, Michigan

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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 ringsIntroducing the ideas behind definitions and theorems to help students develop intuitionHelping students understand how mathematics is done. Students will experiment through examples, make conjectures, and then refine or prove their conjecturesAssisting students in developing their abilities to effectively communicate mathematical ideasActively involving students in realizing each of these goals through in-class and out-of-class activities, common in-class intellectual experiences, and challenging problem setsChanges in the Second EditionStreamlining of introductory material with a quicker transition to the material on rings and groupsNew investigations on extensions of fields and Galois theoryNew exercises added and some sections reworked for clarityMore online Special Topics investigations and additional Appendices, including new appendices on other methods of proof and complex roots of unityEncouraging 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.

Produktinformation

Utgivningsdatum2023-12-19
Mått178 x 254 x 38 mm
Vikt1 152 g
FormatInbunden
SpråkEngelska
SerieTextbooks in Mathematics
Antal sidor525
Upplaga2
FörlagTaylor & Francis Ltd
ISBN9780367555016

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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.

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