Number Theory de Henri Cohen

Number Theory: Graduate Texts in Mathematics, cartea 239

Autor Henri Cohen

en Limba Engleză Paperback – 25 noi 2010

Destinată nivelului de masterat avansat și doctorat, această lucrare semnată de Henri Cohen reprezintă o resursă fundamentală pentru cercetarea în teoria numerelor. Observăm că volumul 240 din prestigioasa serie Graduate Texts in Mathematics a editurii Springer se concentrează pe „teoria explicită a numerelor”, având ca ax central soluționarea ecuațiilor diofantiene în numere întregi, raționale sau algebrice.

Structura operei facilitează o progresie riguroasă: primele capitole sunt dedicate instrumentelor analitice, explorând polinoamele Bernoulli, funcția Gamma și seriile Dirichlet, pentru ca ulterior să introducă perspectiva p-adică. Găsim în a doua parte a cărții instrumente moderne esențiale, precum formele liniare în logaritmi și abordarea modulară, aplicate direct asupra unor probleme celebre, inclusiv ecuația lui Catalan și ecuația Super-Fermat. Comparabil cu The Algorithmic Resolution of Diophantine Equations de Nigel P. Smart în rigoarea matematică, volumul lui Cohen se distinge prin actualizarea metodelor către geometria algebrică aritmetică modernă și prin unificarea subiectului în jurul teoriei funcțiilor zeta și L.

Textul este scris cu o claritate specifică unui autor consacrat, reușind să echilibreze prezentarea teoretică cu aspectul algoritmic. Includerea exercițiilor la finalul fiecărui capitol transformă acest volum dintr-o simplă monografie într-un instrument de lucru activ, necesar oricărui matematician care dorește să stăpânească metodele contemporane de atac asupra punctelor raționale pe curbe de gen superior.

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Specificații

ISBN-13: 9781441923905
ISBN-10: 144192390X
Pagini: 680
Ilustrații: XXIII, 650 p.
Dimensiuni: 155 x 235 x 37 mm
Greutate: 1.01 kg
Ediția:Softcover reprint of hardcover 1st ed. 2007
Editura: Springer
Colecția Graduate Texts in Mathematics
Seria Graduate Texts in Mathematics

Locul publicării:New York, NY, United States

Public țintă

Research

De ce să citești această carte

Această carte se adresează cercetătorilor și studenților la doctorat care au nevoie de o bază solidă în teoria explicită a numerelor. Cititorul câștigă o înțelegere profundă a legăturii dintre funcțiile L și ecuațiile diofantiene, beneficiind de expertiza lui Henri Cohen în metode computaționale și algebrice. Este o resursă indispensabilă pentru cei care studiază geometria aritmetică și caută soluții concrete pentru ecuații polinomiale complexe.

Cuprins

to Diophantine Equations.- to Diophantine Equations.- Tools.- Abelian Groups, Lattices, and Finite Fields.- Basic Algebraic Number Theory.- p-adic Fields.- Quadratic Forms and Local-Global Principles.- Diophantine Equations.- Some Diophantine Equations.- Elliptic Curves.- Diophantine Aspects of Elliptic Curves.

Recenzii

From the reviews:
“The book under review deals with Diophantine analysis from a number-theoretic point of view. … Each chapter ends with exercises, ranging from simple to quite challenging problems. 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.” (Philosophy, Religion and Science Book Reviews, bookinspections.wordpress.com, October, 2013)
“This is the first volume of a highly impressive two-volume textbook on Diophantine analysis. … Readers are presented with an almost overwhelming amount of material. This … text book is bound to become an important reference for students and researchers alike.” (C. Baxa, Monatshefte für Mathematik, Vol. 157 (2), June, 2009)
"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)
"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)

Textul de pe ultima copertă

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 Heegnerpoint 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.

Caracteristici

Unique collection of topics centered around a unifying topic More than 350 exercises Text is largely self-contained

Descriere

Descriere de la o altă ediție sau format:
This book deals with several aspects of what is now called "explicit number theory." The central theme 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 local aspect, global aspect, and the third aspect is the theory of zeta and L-functions. This last aspect can be considered as a unifying theme for the whole subject.