Learning Modern Algebra

A guide to modern algebra for mathematics teachers. It makes explicit connections between abstract algebra and high-school mathematics.

Al Cuoco (Author), Joseph Rotman (Author)

9781939512017, The Mathematical Association of America

Hardback, published 8 January 2015

480 pages
27.5 x 19.5 x 3 cm, 1.08 kg

Much of modern algebra arose from attempts to prove Fermat's Last Theorem, which in turn has its roots in Diophantus' classification of Pythagorean triples. This book, designed for prospective and practising mathematics teachers, makes explicit connections between the ideas of abstract algebra and the mathematics taught at high-school level. Algebraic concepts are presented in historical order, and the book also demonstrates how other important themes in algebra arose from questions related to teaching. The focus is on number theory, polynomials, and commutative rings. Group theory is introduced near the end of the text to explain why generalisations of the quadratic formula do not exist for polynomials of high degree, allowing the reader to appreciate the work of Galois and Abel. Results are motivated with specific examples, and applications range from the theory of repeating decimals to the use of imaginary quadratic fields to construct problems with rational solutions.

Preface
Some features of this book
A note to students
A note to instructors
Notation
1. Early number theory
2. Induction
3. Renaissance
4. Modular arithmetic
5. Abstract algebra
6. Arithmetic of polynomials
7. Quotients, fields, and classical problems
8. Cyclotomic integers
9. Epilogue
References
Index.

Subject Areas: Number theory [PBH], Algebra [PBF], Mathematics [PB]