The central theme of this book is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three of its most basic aspects. The book contains more than 350 exercises and the text is largely self-contained. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five appendices on these techniques.
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From the reviews: "Number Theory, is poised to fill the gap as a core text in number theory … . So, all in all, Henri Cohen’s … Number Theory are, to any mind, an amazing achievement. The coverage is thorough and generally all but encyclopedic, the exercises are good, some are excellent, some will keep even the best-prepared student busy for a long time, and the cultural level of the book … is very high." (Michael Berg, MathDL, July, 2007) "The book under review deals with Diophantine analysis from a number-theoretic point of view. … The clarity of the exposition is the one we expect from the author of two highly successful books on computational number theory … and makes this volume a must-read for researchers in Diophantine analysis." (Franz Lemmermeyer, Zentralblatt MATH, Vol. 1119 (21), 2007) "Cohen (Université Bordeaux I, France), an instant classic, uniquely bridges the gap between old-fashioned, naive treatments and the many modern books available that develop the tools just mentioned … . Summing Up: Recommended. … Upper-division undergraduates through faculty." (D. V. Feldman, CHOICE, Vol. 45 (5), January, 2008)
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The central theme of this graduate-level number theory textbook is the solution of Diophantine equations, i.e., equations or systems of polynomial equations which must be solved in integers, rational numbers or more generally in algebraic numbers. This theme, in particular, is the central motivation for the modern theory of arithmetic algebraic geometry. In this text, this is considered through three aspects. The first is the local aspect: one can do analysis in p-adic fields, and here the author starts by looking at solutions in finite fields, then proceeds to lift these solutions to local solutions using Hensel lifting. The second is the global aspect: the use of number fields, and in particular of class groups and unit groups. This classical subject is here illustrated through a wide range of examples. The third aspect deals with specific classes of equations, and in particular the general and Diophantine study of elliptic curves, including 2 and 3-descent and the Heegner point method. These subjects form the first two parts, forming Volume I. The study of Bernoulli numbers, the gamma function, and zeta and L-functions, and of p-adic analogues is treated at length in the third part of the book, including many interesting and original applications. Much more sophisticated techniques have been brought to bear on the subject of Diophantine equations, and for this reason, the author has included five chapters on these techniques forming the fourth part, which together with the third part forms Volume II. These chapters were written by Yann Bugeaud, Guillaume Hanrot, Maurice Mignotte, Sylvain Duquesne, Samir Siksek, and the author, and contain material on the use of Galois representations, points on higher-genus curves, the superfermat equation, Mihailescu's proof of Catalan's Conjecture, and applications of linear forms in logarithms. The book contains 530 exercises of varying difficulty from immediate consequences of the main text to research problems, and contain many important additional results.
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abelian groups, finite fields, basic algebraic number theory, analytic functions, last theorem, classes modulo squares, modulo cubes, peth power, squarefree integer congruent, double point with distinct tangents, global solubility, minimal monic polynomial, nonconstant isogeny, solution with xyz, nontorsion point, local solubility, quadratic module, ideal coprime, prime ideal decomposition, congruent number problem, only integral solutions, projective solutions, normal basis theorem, general integral solution, pairwise coprime integers Key Phrases - Capitalized Phrases (CAPs): (learn more)
Some Diophantine Equations, Diophantine Equations of Degree, Stickelberger's Theorem, Prove Proposition, Using Proposition, The Quadratic Reciprocity Law, Using Corollary, Field-Theoretic Algebraic Number Theory, Fields Proof, Hilbert's Theorem, The Fermat Quartics, The Normal Basis Theorem, Ring-Theoretic Algebraic Number Theory, Description of General, Consequences of the Hasse-Minkowski Theorem, Basic Results, Catalan's Equation, First Results, Generalizing Corollary, Prove Corollary, Generalizing Proposition, The Hasse-Davenport Relations, Generalize Proposition, Fields Corollary, Using Theorem Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
abelian groups, finite fields, basic algebraic number theory, analytic functions, last theorem, classes modulo squares, modulo cubes, peth power, squarefree integer congruent, double point with distinct tangents, global solubility, minimal monic polynomial, nonconstant isogeny, solution with xyz, nontorsion point, local solubility, quadratic module, ideal coprime, prime ideal decomposition, congruent number problem, only integral solutions, projective solutions, normal basis theorem, general integral solution, pairwise coprime integers Key Phrases - Capitalized Phrases (CAPs): (learn more)
Some Diophantine Equations, Diophantine Equations of Degree, Stickelberger's Theorem, Prove Proposition, Using Proposition, The Quadratic Reciprocity Law, Using Corollary, Field-Theoretic Algebraic Number Theory, Fields Proof, Hilbert's Theorem, The Fermat Quartics, The Normal Basis Theorem, Ring-Theoretic Algebraic Number Theory, Description of General, Consequences of the Hasse-Minkowski Theorem, Basic Results, Catalan's Equation, First Results, Generalizing Corollary, Prove Corollary, Generalizing Proposition, The Hasse-Davenport Relations, Generalize Proposition, Fields Corollary, Using Theorem Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Surprise Me!
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