Integers, Polynomials, and Rings

• Introduction: The McNugget Problem.- Introduction: The McNugget Problem.- Integers.- Induction and the Division Theorem.- The Euclidean Algorithm.- Congruences.- Prime Numbers.- Rings.- Euler’ Theorem.- Binomial Coefficients.- Polynomials.- Polynomials and Roots.- Polynomials with Real Coefficients.- Polynomials with Rational Coefficients.- Polynomial Rings.- Quadratic Polynomials.- Polynomial Congruence Rings.- All Together Now.- Euclidean Rings.- The Ring of Gaussian Integers.- Finite Fields.

From the reviews: "The book focuses mainly on the 'doing' of algebra. ... The chief aim of the author is for students 'to master such skills as learning what a mathematical statement is, what a mathematical argument or proof is, how to present an argument orally ... and how to converse effectively about mathematics.' ... the author strives to motivate students, gradually developing their insights and abilities. ... It is an excellent primer for beginners in the field of abstract algebra, especially for future school teachers." (P. Shiu, The Mathematical Gazette, Vol. 89 (516), 2005) "This is an instructional exposition which treats some elementary number theory ... . It is apparent that the author has made every effort to motivate students resp. to put them in the right way. 'I love algebra. I want my students to love algebra' - I believe that the author succeeded even in this regard." (G. Kowol, Monatshefte fur Mathematik, Vol. 144 (2), 2005) "This is a very elementary introduction to elementary number theory and some related topics in algebra ... . The topics chosen are well suited for a student's first exposure to 'serious' mathematics (much more so, in the reviewer's opinion, than the calculus course that is the norm in almost all curricula almost everywhere)." (S. Frisch, Internationale Mathematische Nachrichten, Issue 196, 2004) "The book ... represents a very special introduction to modern algebra ... . focuses less on contents and more on the 'doing' of algebra. ... Many proofs are left as exercises, together with detailed hints or outlines, and these exercises actually form the heart of the entire text. ... Summing up, this book is a great primer for beginners in the field ... . could serve well in an undergraduate course for non-mathematicians, and as a guide to self-education beyond academic training, too." (Werner Kleinert, Zentralblatt MATH, Vol. 1046 (2), 2004) "Originally conceived as a text for a course for future secondary-school mathematics teachers, this book has developed into one that could serve well in an undergraduate course in abstract algebra ... . The topics studies should be of interest to all mathematics students and are especially appropriate for future teachers. ... Many proofs are left as exercises, and for almost every such exercise, a detailed hint or outline of the proof is provided. These exercises form the heart of the text." (Zentralblatt fur Didaktik der Mathematik, November, 2004) "Mathematics is often regarded as the study of calculation ... . It combines creativity and logic in order to arrive at abstract truths. This book is intended to illustrate how calculation, creativity, and logic can be combined to solve a range of problems in algebra. ... Many proofs are left as exercises, and for almost every such exercise, a detailed hint or outline of the proof is provided. These exercises form the heart of the text." (L'Enseignement Mathematique, Vol. 50 (1-2), 2004) "The book is meant to be a structurally different abstract algebra textbook. ... the book is very unitary and it has a good flow. ... Integers, Polynominals and Rings is a unique book, and should be extremely useful for an audience of future high school teachers. It would also be a valuable supplement for students taking a traditional abstract algebra course, especially since it is very readable." (Ioana Mihaila, MathDL, January, 2004)