Algebraic Number TheoryPublisher Description (unedited publisher data) This is a corrected printing of the second edition of Lang's well-known textbook. It covers all of the basic material of classical algebraic number theory, giving the student the background necessary for the study of further topics in algebraic number theory, such as cyclotomic fields, or modular forms. Part I introduces some of the basic ideas of the theory: number fields, ideal classes, ideles and adeles, and zeta functions. It also contains a version of a Riemann-Roch theorem in number fields, proved by Lang in the very first version of the book in the sixties. This version can now be seen as a precursor of Arakelov theory. Part II covers class field theory, and Part III is devoted to analytic methods, including an exposition of Tate's thesis, the Brauer-Siegel theorem, and Weil's explicit formulas. The second edition contains corrections, as well as several additions to the previous edition, and the last chapter on explicit formulas has been rewritten. |
Contents
CHAPTER I | 3 |
Chinese remainder theorem | 11 |
Dedekind rings | 21 |
Copyright | |
75 other sections not shown
Other editions - View all
Common terms and phrases
a₁ abelian extension adele apply archimedean absolute values Artin map assertion assume automorphism b₁ bounded Chapter character class field theory class group coefficients compact complex numbers compute concludes the proof constant contained converges Corollary cyclic decomposition group Dedekind ring define denote Dirichlet series discriminant element embedded equal exists fact finite extension finite number follows formula Fourier transform fractional ideal function f functional equation fundamental domain Galois group group G Haar measure Hence homomorphism ideal class ideal class group idele inequality integral closure isomorphism kernel L-series lattice Lemma Let f Let K/k log Np maximal ideal module non-zero norm notation number field open subgroup p-adic polynomial prime ideal prime number Proposition prove quasi-character quotient field r₁ ramified Re(s roots of unity satisfies sin² splits completely subset Tauberian theorem trivial unit unramified variables vector zeros zeta function