000 04143nam a22005415i 4500
001 978-0-387-49923-9
003 DE-He213
005 20161121231123.0
007 cr nn 008mamaa
008 100301s2007 xxu| s |||| 0|eng d
020 _a9780387499239
_9978-0-387-49923-9
024 7 _a10.1007/978-0-387-49923-9
_2doi
050 4 _aQA241-247.5
072 7 _aPBH
_2bicssc
072 7 _aMAT022000
_2bisacsh
082 0 4 _a512.7
_223
100 1 _aCohen, Henri.
_eauthor.
245 1 0 _aNumber Theory
_h[electronic resource] :
_bVolume I: Tools and Diophantine Equations /
_cby Henri Cohen.
264 1 _aNew York, NY :
_bSpringer New York,
_c2007.
300 _aXXIII, 650 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aGraduate Texts in Mathematics,
_x0072-5285 ;
_v239
505 0 _ato 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.
520 _aThe 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.
650 0 _aMathematics.
650 0 _aAlgebra.
650 0 _aField theory (Physics).
650 0 _aOrdered algebraic structures.
650 0 _aComputer mathematics.
650 0 _aAlgorithms.
650 0 _aNumber theory.
650 1 4 _aMathematics.
650 2 4 _aNumber Theory.
650 2 4 _aAlgorithms.
650 2 4 _aField Theory and Polynomials.
650 2 4 _aComputational Mathematics and Numerical Analysis.
650 2 4 _aOrder, Lattices, Ordered Algebraic Structures.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9780387499222
830 0 _aGraduate Texts in Mathematics,
_x0072-5285 ;
_v239
856 4 0 _uhttp://dx.doi.org/10.1007/978-0-387-49923-9
912 _aZDB-2-SMA
950 _aMathematics and Statistics (Springer-11649)
999 _c508900
_d508900