Algebraic Theory of Numbers

ISBN-10: 0486466663

ISBN-13: 9780486466668

Edition: 2008

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Description: Algebraic number theory introduces studentsnbsp;to new algebraic notions as well asnbsp;related concepts: groups, rings, fields, ideals, quotient rings, and quotient fields. This text covers the basics, from divisibility theory in principal ideal domains to the unit theorem, finiteness of the class number, and Hilbert ramification theory. 1970 edition.

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Book details

List price: $10.95
Copyright year: 2008
Publisher: Dover Publications, Incorporated
Publication date: 5/19/2008
Binding: Paperback
Pages: 112
Size: 6.00" wide x 9.25" long x 0.50" tall
Weight: 0.352
Language: English

Translator's Introduction
Introduction
Notations, Definitions, and Prerequisites
Principal ideal rings
Divisibility in principal ideal rings
An example: the diophantine equations X[superscript 2] + Y[superscript 2] = Z[superscript 2] and X[superscript 4] + Y[superscript 4] = Z[superscript 4]
Some lemmas concerning ideals; Euler's [characters not reproducible]-function
Some preliminaries concerning modules
Modules over principal ideal rings
Roots of unity in a field
Finite fields
Elements integral over a ring; elements algebraic over a field
Elements integral over a ring
Integrally closed rings
Elements algebraic over a field. Algebraic extensions
Conjugate elements, conjugate fields
Integers in quadratic fields
Norms and traces
The discriminant
The terminology of number fields
Cyclotomic fields
The field of complex numbers is algebraically closed
Noetherian rings and Dedekind rings
Noetherian rings and modules
An application concerning integral elements
Some preliminaries concerning ideals
Dedekind rings
The norm of an ideal
Ideal classes and the unit theorem
Preliminaries concerning discrete subgroups of R[superscript n]
The canonical imbedding of a number field
Finiteness of the ideal class group
The unit theorem
Units in imaginary quadratic fields
Units in real quadratic fields
A generalization of the unit theorem
The calculation of a volume
The splitting of prime ideals in an extension field
Preliminaries concerning rings of fractions
The splitting of a prime ideal in an extension
The discriminant and ramification
The splitting of a prime number in a quadratic field
The quadratic reciprocity law
The two-squares theorem
The four-squares theorem
Galois extensions of number fields
Galois theory
The decomposition and inertia groups
The number field case. The Frobenius automorphism
An application to cyclotomic fields
Another proof of the quadratic reciprocity law
A Supplement, Without Proofs
Exercises
Bibliography
Index
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