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

__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. front cover ......Page 1 copyright page ......Page 3 title page ......Page 4 Contents......Page 10 Preface......Page 14 Some Features of This Book......Page 15 A Note to Instructors......Page 16 Notation......Page 18 Ancient Mathematics......Page 22 Diophantus......Page 28 Geometry and Pythagorean Triples......Page 29 The Method of Diophantus......Page 32 Fermat's Last Theorem......Page 35 Connections: Congruent Numbers......Page 37 Euclid......Page 41 Greek Number Theory......Page 42 Division and Remainders......Page 43 Linear Combinations and Euclid's Lemma......Page 45 Euclidean Algorithm......Page 51 Nine Fundamental Properties......Page 57 Trigonometry......Page 62 Integration......Page 63 Induction and Applications......Page 66 Unique Factorization......Page 74 Strong Induction......Page 78 Differential Equations......Page 81 Binomial Theorem......Page 84 Combinatorics......Page 90 An Approach to Induction......Page 94 Fibonacci Sequence......Page 96 Renaissance......Page 102 Classical Formulas......Page 103 Complex Numbers......Page 112 Algebraic Operations......Page 114 Absolute Value and Direction......Page 120 The Geometry Behind Multiplication......Page 122 Roots and Powers......Page 127 Connections: Designing Good Problems......Page 137 Norms......Page 138 Pippins and Cheese......Page 139 Gaussian Integers: Pythagorean Triples Revisited......Page 140 Eisenstein Triples and Diophantus......Page 143 Nice Boxes......Page 144 Nice Functions for Calculus Problems......Page 145 Lattice Point Triangles......Page 147 Congruence......Page 152 Public Key Codes......Page 170 Commutative Rings......Page 175 Units and Fields......Page 181 Subrings and Subfields......Page 187 Connections: Julius and Gregory......Page 190 Real Numbers......Page 198 Decimal Expansions of Rationals......Page 200 Periods and Blocks......Page 203 Abstract Algebra......Page 212 Domains and Fraction Fields......Page 213 Polynomials......Page 217 Polynomial Functions......Page 225 Homomorphisms......Page 227 Extensions of Homomorphisms......Page 234 Kernel, Image, and Ideals......Page 237 Connections: Boolean Things......Page 242 Inclusion-Exclusion......Page 248 Divisibility......Page 254 Roots......Page 260 Greatest Common Divisors......Page 265 Unique Factorization......Page 269 Principal Ideal Domains......Page 276 Irreducibility......Page 280 Roots of Unity......Page 285 Connections: Lagrange Interpolation......Page 291 Quotient Rings......Page 298 Characteristics......Page 308 Extension Fields......Page 310 Algebraic Extensions......Page 314 Splitting Fields......Page 321 Classification of Finite Fields......Page 326 Connections: Ruler--Compass Constructions......Page 329 Constructing Regular n-gons......Page 342 Gauss's construction of the 17-gon......Page 344 Cyclotomic Integers......Page 350 Arithmetic in Gaussian and Eisenstein Integers......Page 351 Euclidean Domains......Page 355 Primes Upstairs and Primes Downstairs......Page 358 Laws of Decomposition......Page 360 Fermat's Last Theorem for Exponent 3......Page 370 Preliminaries......Page 371 The First Case......Page 372 Gauss's Proof of the Second Case......Page 375 Approaches to the General Case......Page 380 Cyclotomic integers......Page 381 Kummer, Ideal Numbers, and Dedekind......Page 386 Connections: Counting Sums of Squares......Page 392 A Proof of Fermat's Theorem on Divisors......Page 394 Abel and Galois......Page 400 Solvability by Radicals......Page 402 Symmetry......Page 405 Groups......Page 410 Wiles and Fermat's Last Theorem......Page 417 Elliptic Integrals and Elliptic Functions......Page 418 Congruent Numbers Revisited......Page 421 Elliptic Curves......Page 425 Functions......Page 430 Equivalence Relations......Page 441 Vector Spaces......Page 445 Bases and Dimension......Page 448 Linear Transformations......Page 456 Inequalities......Page 462 Generalized Associativity......Page 463 A Cyclotomic Integer Calculator......Page 465 Eisenstein Integers......Page 466 Algebra with Periods......Page 467 References......Page 470 Index......Page 472 About the Authors......Page 480 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. Learning Modern Algebra Is Designed For College Students Who Want To Teach Mathematics In High School, 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.--publisher Description. Early Number Theory -- Induction -- Renaissance -- Modular Arithmetic -- Abstract Algebra -- Arithmetic Of Polynomials -- Quotients, Fields, And Classical Problems -- Cyclotomic Integers -- Epilog -- Appendices. Al Cuoco And Joseph J. Rotman. Includes Bibliographical References (p. 449-450) And Index. "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 Phythagorean 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 arouse from questions related to teaching"--P. [4] of cover.