rdf-syntax-grammar
mods:
  schema location: http://www.loc.gov/mods/v3 http://www.loc.gov/standards/mods/v3/mods-3-2.xsd
  title info:
    title: Number Theory
    sub title: Volume I: Tools and Diophantine Equations
  name:
    name part: Cohen, Henri.
    role:
      role term: creator
    role:
      role term: author.
  name:
    name part: SpringerLink (Online service)
  type of resource: text
  origin info:
    place:
      place term: xxu
    date issued: 2007
    issuance: monographic
  language:
    language term: eng
  physical description:
    form: electronic
    form: electronic resource
    reformatting quality: access
    form: remote
    extent: XXIII, 650 p. online resource.
  abstract: 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 m…
  table of contents: 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.
  note: by Henri Cohen.
  subject:
    topic: Mathematics
  subject:
    topic: Algebra
  subject:
    topic: Field theory (Physics)
  subject:
    topic: Ordered algebraic structures
  subject:
    topic: Computer mathematics
  subject:
    topic: Algorithms
  subject:
    topic: Number theory
  subject:
    topic: Mathematics
  subject:
    topic: Number Theory
  subject:
    topic: Algorithms
  subject:
    topic: Field Theory and Polynomials
  subject:
    topic: Computational Mathematics and Numerical Analysis
  subject:
    topic: Order, Lattices, Ordered Algebraic Structures
  classification: QA241-247.5
  classification: 512.7
  related item:
    title info:
      title: Springer eBooks
  related item:
    title info:
      title: Graduate Texts in Mathematics, 239
  identifier: 9780387499239
  identifier: http://dx.doi.org/10.1007/978-0-387-49923-9
  record info:
    record creation date: 100301
    record change date: 20151204190207.0
    record identifier: 978-0-387-49923-9