Algebraic Number Theory

  title={Algebraic Number Theory},
  author={J{\"u}rgen Neukirch},
I: Algebraic Integers.- II: The Theory of Valuations.- III: Riemann-Roch Theory.- IV: Abstract Class Field Theory.- V: Local Class Field Theory.- VI: Global Class Field Theory.- VII: Zeta Functions and L-series. 
Geometry of Numbers
We develop a global cohomology theory for number fields by offering topological cohomology groups, an arithmetical duality, a Riemann-Roch type theorem, and two types of vanishing theorem. As
The Interplay of Invariant Theory with Multiplicative Ideal Theory and with Arithmetic Combinatorics
This paper surveys and develops links between polynomial invariants of finite groups, factorization theory of Krull domains, and product-one sequences over finite groups. The goal is to gain a better
Arithmetic on Groups of Positive Rationals
Abstract In this paper we develop a theory of unique factorization for subgroups of the positive rationals. We show that this theory is strong enough to include arithmetic progressions and the theory
Number Theory and Polynomials: The Mahler measure of algebraic numbers: a survey
A survey of results for Mahler measure of algebraic numbers, and one-variable polynomials with integer coefficients is presented. Related results on the maximum modulus of the conjugates (`house') of
K-theory for ring c*-algebras: the case of number fields with higher roots of unity
We compute K-theory for ring C*-algebras in the case of higher roots of unity and thereby completely determine the K-theory for ring C*-algebras attached to rings of integers in arbitrary number
This work is a historical exposition of mathematical ideas, methods and research programs which supported the birth and growth of modern Algebraic Number Theory. The mathematicians picked up here are
Elliptic Curves and the Mordell-Weil Theorem
This paper introduces the notion of elliptic curves with an emphasis on elliptic curves defined over Q and their rational points. Some algebraic number theory and algebraic geometry is developed in
In this paper, we aim to study valuations on finite extensions of Q. These extensions fall under a special type of field called a global field. We shall also cover the topics of the Approximation
Arithmetic theory of q-difference equations. The q-analogue of Grothendieck-Katz's conjecture on p-curvatures
Grothendieck's conjecture on p-curvatures predicts that an arithmetic differential equation has a full set of algebraic solutions if and only if its reduction in positive characteristic has a full


A Course in Arithmetic
Part 1 Algebraic methods: finite fields p-adic fields Hilbert symbol quadratic forms over Qp, and over Q integral quadratic forms with discriminant +-1. Part 2 Analytic methods: the theorem on
The arithmetic of elliptic curves
  • J. Silverman
  • Mathematics, Computer Science
    Graduate texts in mathematics
  • 1986
It is shown here how Elliptic Curves over Finite Fields, Local Fields, and Global Fields affect the geometry of the elliptic curves.
Algebraic Number Theory
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
Advanced Topics in the Arithmetic of Elliptic Curves
In The Arithmetic of Elliptic Curves, the author presented the basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rational
An Introduction to Homological Algebra
An Introduction to Homological Algebra discusses the origins of algebraic topology. It also presents the study of homological algebra as a two-stage affair. First, one must learn the language of Ext
  • Computer Science
    Brain Research Bulletin
  • 1986
Homological Algebra (PMS-19)