Learning Modern Algebra : From Early Attempts to Prove Fermat's Last Theorem.

front cover -- copyright page -- title page -- Contents -- Preface -- Some Features of This Book -- A Note to Students -- A Note to Instructors -- Notation -- Early Number Theory -- Ancient Mathematics -- Diophantus -- Geometry and Pythagorean Triples -- The Method of Diophantus -- Fermat's Last Theorem -- Connections: Congruent Numbers -- Euclid -- Greek Number Theory -- Division and Remainders -- Linear Combinations and Euclid's Lemma -- Euclidean Algorithm -- Nine Fundamental Properties -- Connections -- Trigonometry -- Integration -- Induction -- Induction and Applications -- Unique Factorization -- Strong Induction -- Differential Equations -- Binomial Theorem -- Combinatorics -- Connections -- An Approach to Induction -- Fibonacci Sequence -- Renaissance -- Classical Formulas -- Complex Numbers -- Algebraic Operations -- Absolute Value and Direction -- The Geometry Behind Multiplication -- Roots and Powers -- Connections: Designing Good Problems -- Norms -- Pippins and Cheese -- Gaussian Integers: Pythagorean Triples Revisited -- Eisenstein Triples and Diophantus -- Nice Boxes -- Nice Functions for Calculus Problems -- Lattice Point Triangles -- Modular Arithmetic -- Congruence -- Public Key Codes -- Commutative Rings -- Units and Fields -- Subrings and Subfields -- Connections: Julius and Gregory -- Connections: Patterns in Decimal Expansions -- Real Numbers -- Decimal Expansions of Rationals -- Periods and Blocks -- Abstract Algebra -- Domains and Fraction Fields -- Polynomials -- Polynomial Functions -- Homomorphisms -- Extensions of Homomorphisms -- Kernel, Image, and Ideals -- Connections: Boolean Things -- Inclusion-Exclusion -- Arithmetic of Polynomials -- Parallels to Z -- Divisibility -- Roots -- Greatest Common Divisors -- Unique Factorization -- Principal Ideal Domains -- Irreducibility -- Roots of Unity.

Connections: Lagrange Interpolation -- Quotients, Fields, and Classical Problems -- Quotient Rings -- Field Theory -- Characteristics -- Extension Fields -- Algebraic Extensions -- Splitting Fields -- Classification of Finite Fields -- Connections: Ruler--Compass Constructions -- Constructing Regular n-gons -- Gauss's construction of the 17-gon -- Cyclotomic Integers -- Arithmetic in Gaussian and Eisenstein Integers -- Euclidean Domains -- Primes Upstairs and Primes Downstairs -- Laws of Decomposition -- Fermat's Last Theorem for Exponent 3 -- Preliminaries -- The First Case -- Gauss's Proof of the Second Case -- Approaches to the General Case -- Cyclotomic integers -- Kummer, Ideal Numbers, and Dedekind -- Connections: Counting Sums of Squares -- A Proof of Fermat's Theorem on Divisors -- Epilog -- Abel and Galois -- Solvability by Radicals -- Symmetry -- Groups -- Wiles and Fermat's Last Theorem -- Elliptic Integrals and Elliptic Functions -- Congruent Numbers Revisited -- Elliptic Curves -- Appendices -- Functions -- Equivalence Relations -- Vector Spaces -- Bases and Dimension -- Linear Transformations -- Inequalities -- Generalized Associativity -- A Cyclotomic Integer Calculator -- Eisenstein Integers -- Symmetric Polynomials -- Algebra with Periods -- References -- Index -- About the Authors.

Learning Modern Algebra aligns with the CBMS Mathematical Education of Teachers II recommendations, in both content and practice. It emphasizes rings and fields over groups, and it makes explicit connections between the ideas of abstract algebra and the mathematics used by high school teachers. It provides opportunities for prospective and practicing teachers to experience mathematics for themselves, before the formalities are developed, and it is explicit about the mathematical habits of mind that lie beneath the definitions and theorems.This book is designed for prospective and practicing high school mathematics teachers, but it can serve as a text for standard abstract algebra courses as well. The presentation is organized historically: the Babylonians introduced Pythagorean triples to teach the Pythagorean theorem; these were classified by Diophantus, and eventually this led Fermat to conjecture his Last Theorem. The text shows how much of modern algebra arose in attempts to prove this; it also shows how other important themes in algebra arose from questions related to teaching. Indeed, modern algebra is a very useful tool for teachers, with deep connections to the actual content of high school mathematics, as well as to the mathematics teachers use in their profession that doesn't necessarily "end up on the blackboard."The focus is on number theory, polynomials, and commutative rings. Group theory is introduced near the end of the text to explain why generalizations of the quadratic formula do not exist for polynomials of high degree, allowing the reader to appreciate the more general work of Galois and Abel on roots of polynomials. Results and proofs are motivated with specific examples whenever possible, so that abstractions emerge from concrete experience. Applications range from the theory of repeating decimals to the use of imaginary

quadratic fields to construct problems with rational solutions. While such applications are integrated throughout, each chapter also contains a section giving explicit connections between the content of the chapter and high school teaching.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.