Algebraic Number Fields

This text presents the basic information about finite dimensional extension fields of the rational numbers, algebraic number fields, and the rings of algebraic integers in them. The important theorems regarding the units of the ring of integers and the class group are proved and illustrated with many examples given in detail. The completion of an algebraic number field at a valuation is discussed in detail and then used to provide economical proofs of global results. The book contains many concrete examples illustrating the computation of class groups, class numbers, and Hilbert class fields. Exercises are provided to indicate applications of the general theory.

A translation of Hilberts "Theorie der algebraischen Zahlkörper" best known as the "Zahlbericht", first published in 1897, in which he provides an elegantly integrated overview of the development of algebraic number theory up to the end of the nineteenth century. The Zahlbericht also provided a firm foundation for further research in the theory, and can be seen as the starting point for all twentieth century investigations into the subject, as well as reciprocity laws and class field theory. This English edition further contains an introduction by F. Lemmermeyer and N. Schappacher.

Requiring no more than a basic knowledge of abstract algebra, this text presents the mathematics of number fields in a straightforward, pedestrian manner. It therefore avoids local methods and presents proofs in a way that highlights the important parts of the arguments. Readers are assumed to be able to fill in the details, which in many places are left as exercises.

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"Artin's 1932 Göttingen Lectures on Class Field Theory" and "Connections between Algebrac Number Theory and Integral Matrices"

Broad graduate-level account of Algebraic Number Theory, first published in 2001, including exercises, by a world-renowned author.

This second edition is a corrected and extended version of the first. It is a textbook for students, as well as a reference book for the working mathematician, on cohomological topics in number theory. In all it is a virtually complete treatment of a vast array of central topics in algebraic number theory. New material is introduced here on duality theorems for unramified and tamely ramified extensions as well as a careful analysis of 2-extensions of real number fields.

Early in the development of number theory, it was noticed that the ring of integers has many properties in common with the ring of polynomials over a finite field. The first part of this book illustrates this relationship by presenting analogues of various theorems. The later chapters probe the analogy between global function fields and algebraic number fields. Topics include the ABC-conjecture, Brumer-Stark conjecture, and Drinfeld modules.