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1999 | Buch

Algebraic Integers Kapitel lesen Erstes Kapitel lesen

Algebraic Number Theory

verfasst von: Jürgen Neukirch

Verlag: Springer Berlin Heidelberg

Buchreihe : Grundlehren der mathematischen Wissenschaften

Enthalten in: Professional Book Archive

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Über dieses Buch

From the review: "The present book has as its aim to resolve a discrepancy in the textbook literature and ... to provide a comprehensive introduction to algebraic number theory which is largely based on the modern, unifying conception of (one-dimensional) arithmetic algebraic geometry. ... Despite this exacting program, the book remains an introduction to algebraic number theory for the beginner... The author discusses the classical concepts from the viewpoint of Arakelov theory.... The treatment of class field theory is ... particularly rich in illustrating complements, hints for further study, and concrete examples.... The concluding chapter VII on zeta-functions and L-series is another outstanding advantage of the present textbook.... The book is, without any doubt, the most up-to-date, systematic, and theoretically comprehensive textbook on algebraic number field theory available."
W. Kleinert in: Zentralblatt für Mathematik, 1992

Inhaltsverzeichnis

Frontmatter
Chapter I. Algebraic Integers
Abstract
The equations
$${\text{2 = 1 + 1,5 = 1 + 4,13 = 4 + 9,17 = 1 + 16,29 = 4 + 25,37 = 1 + 36}}$$
show the first prime numbers that can be represented as a sum of two squares. Except for 2, they are all ≡ 1 mod 4, and it is true in general that any odd prime number of the form p = a 2 + b 2 satisfies p ≡ 1 mod 4, because perfect squares are ≡ 0 or ≡ 1 mod 4. This is obvious.
Jürgen Neukirch
Chapter II. The Theory of Valuations
Abstract
The p-adic numbers were invented at the beginning of the twentieth century by the mathematician Kurt Hensel (1861–1941) with a view to introduce into number theory the powerful method of power series expansion which plays such a predominant rôle in function theory. The idea originated from the observation made in the last chapter that the numbers f ∈ ℤ may be viewed in analogy with the polynomials f (z) ∈ ℂ[z] as functions on the space X of prime numbers in ℤ, associating to them their “value” at the point p ∈ X, i.e., the element
$$f\left( p \right): = f\,\bmod \,p$$
in the residue class field k(p) = ℤ/pℤ.
Jürgen Neukirch
Chapter III. Riemann-Roch Theory
Abstract
Having set up the general theory of valued fields, we now return to algebraic number fields. We want to develop their basic theory from the valuation-theoretic point of view. This approach has two prominent advantages compared to the ideal-theoretic treatment given in the first chapter. The first one results from the possibility of passing to completions, thereby enacting the important “local-to-global principle”. This will be done in chapter VI. The other advantage lies in the fact that the archimedean valuations bring into the picture the points at infinity, which were hitherto lacking, as the “primes at infinity”. In this way a perfect analogy with the function fields is achieved. This is the task to which we now turn.
Jürgen Neukirch
Chapter IV. Abstract Class Field Theory
Abstract
Every field k is equipped with a distinguished Galois extension: the separable closure \(\bar k|k\). Its Galois group \({G_k} = G(\bar k|k)\) is called the absolute Galois group of k. As a rule, this extension will have infinite degree. It does, however, have the advantage of collecting all finite Galois extensions of k. This is why it is reasonable to try to give it a prominent place in Galois theory. But such an attempt faces the difficulty that the main theorem of Galois theory does not remain true for infinite extensions. Let us explain this in the following
Jürgen Neukirch
Chapter V. Local Class Field Theory
Abstract
The abstract class field theory that we have developed in the last chapter is now going to be applied to the case of a local field, i.e., to a field which is complete with respect to a discrete valuation, and which has a finite residue class field. By chap. II, (5.2), these are precisely the finite extensions K of the fields ℚ p or F p ((t)). We will use the following notation. Let
  • υ K be the discrete valuation normalized by υ K (K*) = ℤ,
  • O K = {a ∈ K | υ K (a) ≥ 0} the valuation ring,
  • p K = {a ∈ K | υ K (a) > 0} the maximal ideal,
  • κ = O K /p K the residue class field,
  • U K = {a ∈ K* | υ K (a) = 0} the unit group,
  • U K (n) = 1 + p K n the group of n-th higher units, n = 1, 2, ... ,
  • q = q K = #κ,
  • |a|p = q −υ K (a) the normalized p-adic absolute value,
  • μ n the group of n-th roots of unity, and μ n (K) = μ n ∩ K*.
  • π K , or simply π, denotes a prime element of K, i.e., p K = πO K .
Jürgen Neukirch
Chapter VI. Global Class Field Theory
Abstract
The rôle held in local class field theory by the multiplicative group of the base field is taken in global class field theory by the idèle class group. The notion of idèle is a modification of the notion of ideal. It was introduced by the French mathematician Claude Chevalley (1909–1984) with a view to providing a suitable basis for the important local-to-global principle, i.e., for the principle which reduces problems concerning a number field K to analogous problems for the various completions K p . Chevalley used the term “ideal element”, which was abbreviated as id. el.
Jürgen Neukirch
Chapter VII. Zeta Functions and L-series
Abstract
One of the most astounding phenomena in number theory consists in the fact that a great number of deep arithmetic properties of a number field are hidden within a single analytic function, its zeta function. This function has a simple shape, but it is unwilling to yield its mysteries. Each time, however, that we succeed in stealing one of these well-guarded truths, we may expect to be rewarded by the revelation of some surprising and significant relationship. This is why zeta functions, as well as their generalizations, the L-series, have increasingly moved to the foreground of the arithmetic scene, and today are more than ever the focus of number-theoretic research.
Jürgen Neukirch
Backmatter
Metadaten
Titel
Algebraic Number Theory
verfasst von
Jürgen Neukirch
Copyright-Jahr
1999
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-662-03983-0
Print ISBN
978-3-642-08473-7
DOI
https://doi.org/10.1007/978-3-662-03983-0