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Analysis and Algebra is intended for use in introductory real analysis and abstract algebra courses. The book explores the standard notions in both areas with an expressed goal of highlighting the connections between analysis and algebra. The exposition of the material, together with selected proofs and numerous exercises, make the book very student-friendly. Included are an entirely analytic proof of the Fundamental Theorem of Algebra and an entirely algebraic proof of an interpolation theorem in real analysis. To encourage student participation in the learning process, most results do not include their proofs; rather, the students will be called upon to construct the proofs of the results in guided exercises.
Each chapter begins with a "Getting Your Hands Dirty" section comprised of a few exercises. The goal of these problems is to reinforce the material presented in the chapter and to open avenues of inquiry. As such, these questions may prove challenging, but students should be able to answer them after the completion of each chapter. Several of the chapters also contain a section on the history of the concepts and problems under discussion. These sections are intended to "bring mathematics alive" and place its development in perspective. In fact, it is our hope that students will find their curiosity only aroused and not satisfied, so that they continue to explore mathematics on their own. Finally, the hundreds of exercises throughout the text aim to promote active learning and contain several key results which we call upon throughout this book. Therefore each exercise should be at least attempted if not completed in the interest of a full learning experience.
Dr. Oscar Vega is an Assistant Professor in the mathematics department at California State University, Fresno. He received his PhD in Mathematics in 2006 from the University of Iowa.
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