Solutions Manual An Introduction to Abstract Algebra with Notes to the Future Teacher

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Solutions Manual An Introduction to Abstract Algebra with Notes to the Future Teacher

This traditional treatment of abstract algebra is designed for the particular needs of the mathematics teacher. Readers must have access to a Computer Algebra System (C. A. S.) such as Maple, or at minimum a calculator such as the TI 89 with C. A. S. capabilities. Includes “To the Teacher” sections that Draw connections from the number theory or abstract algebra under consideration to secondary mathematics. Provides historical context with “From the Past” sections in each chapter. Features “Worksheets” that outline the framework of a topic in most chapters. A useful reference for mathematics teachers who need to brush up on their abstract algebra skills.

Despite its awkward title, An Introduction to Abstract Algebra with Notes to the Future Teacher is exactly the kind of text that math teacher training programs need. Nicodemi, Sutherland, and Towsley provide a solid, rigorous introduction to abstract algebra, but at the same time help future teachers connect the subject to the high school curriculum.

Most sections of this book end with a note “To the Teacher” and each chapter has a section called “In the Classroom,” in which the text addresses aspiring teachers in two main ways. First, if topics come up that are appropriate for the high school classroom, the authors “point these out and point to how they might be implemented.” For example, after a section on primes and unique factorization, Nicodemi, Sutherland, and Towsley encourage future teachers to challenge a class with a lesson about the Twin Prime Conjecture and the Goldbach Conjecture, as they are ideas that intrigue mathematicians and curious students alike.

Second, and perhaps even more important, by studying more advanced topics in abstract algebra, the future teacher gains a “deeper knowledge that will allow you to see the high school curriculum in larger context.” One example of this is when the authors examine the typical high school algebra topic of factoring completely, examining the meaning of this in different rings. Many textbooks are vague in their directions to the students, and usually expect the student to know that they are factoring over the ring of integers. If factoring x4 – 25, Nicodemi, Sutherland, and Towsley show that the answers are distinct if factoring over integers, real numbers, or complex numbers. The text helps prospective teachers to clear up these ideas in their own minds, so that they can explain it more clearly to their students.

While these connections to the familiar territory of high school algebra will obviously serve the future teacher, they will also enhance all learners’ understanding of abstract algebra. In other words, the text is, simply put, a well-written introductory abstract algebra text, regardless of whether the reader wants to be a high school math teacher. As an introductory text, its use of numbers and concrete examples before moving to abstract theorems will definitely benefit the beginning math student. For instance, before proving Euler’s Theorem, the authors work through the steps of the proof with numbers, thus “anticipating the steps in the proof that follows.”

In accordance with its introductory nature, the book spends some developing the basics in number theory (like the division algorithm and modular arithmetic) before moving onto rings, groups, and fields. Still, it is rigorous enough to reach the unsolvability of the general fifth degree polynomial in the final section.

Another aspect of the text which is simple yet so helpful is the chapter summaries and chapter exercises. They highlight the main points of the chapter and allow the student to check how well they understood the chapter. The authors provide thoughtful homework problems: simple calculations, open-ended questions, and classic ones, like an ancient Chinese problem about a band of 17 pirates divvying up coins.

Nicodemi, Sutherland, and Towsley frequently make connections with theorems learned earlier in the text, so that the reader can visit and revisit ideas, gaining the ability to see one idea from multiple vantage points. For example, first the reader learns about the expressing the greatest common multiple of two integers as an equation. Later they connect this to solving Diophantine equations, then to solving modular equations, and finally to units in Z mod m.

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