This book details the classical part of the theory of algebraic number theory, excluding class-field theory and its consequences. Coverage includes: ideal theory in rings of algebraic integers, p-adic fields and their finite extensions, ideles and adeles, zeta-functions, distribution of prime ideals, Abelian fields, the class-number of quadratic fields, and factorization problems. The book also features exercises and a list of open problems.
Elementary and Analytic Theory of Algebraic Numbers
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Editorial Reviews
Review
From the reviews of the third edition: "This giant tome is ‘elementary’ only in the sense of ‘classical’ (for the first four chapters), and ‘analytic’ thereafter … . The main text of nine chapters has about 400 pages. There are extensive notes at the end of each chapter, covering a further 90 pages and giving additional background with references to the research literature. … is likely to be, for many years, a valuable resource for those already involved in serious research on algebraic number theory … ." (John Baylis, The Mathematical Gazette, Vol. 89 (515), 2005) "The book gives an exposition of the classical part of the theory of algebraic number theory, excluding class-field theory and its consequences. … Each chapter ends with exercises and a short review of the relevant literature up to 2003. The bibliography has over 3400 items." (Zentralblatt für Didaktik der Mathematik, November, 2004) "This is the third edition of a well-known textbook on algebraic number theory. … the author has thoroughly updated the comments at the end of each chapter and extended the bibliography accordingly. … for most professional number theorists one of this book’s most appealing features will be the carefully researched notes and the large number of references. These make it a real treasure house and ensure that this volume will be an indispensable work of reference for anyone working in or using algebraic number theory." (Ch. Baxa, Monatshefte für Mathematik, Vol. 149 (2), 2006) "Narkiewicz’ tome, weighing in at xi + 708 pages, is an imposing work, ambitious in scope and even encyclopaedic … . Elementary and Analytic Theory of Algebraic Numbers is also well-written and eminently readable by a good and diligent graduate student. It would serve beautifully for a graduate-level course in number theory sans class-field theory. … Narkiewicz’ presentation is so clear and detailed that coverage of certain topics … is extremely beneficial." (Michael Berg, MathDL, September, 2005) "It is a very great pleasure to have the third edition at hand … . the book has already been a Bible now for generations of number theorists. … if someone has his special interest in algebraic number theory, analytic number theory, Diophantine equations or whatever: this book has always turned out to be crucial. … There is a list of Problems … all of which represent a current research direction of number theory." (István Gaál, Zentralblatt MATH, Vol. 1159, 2009)
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Inside This Book
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First Sentence:
1. This chapter is introductory, and contains the fundamental properties of Dedekind domains including their behaviour under finite extensions and the structure of finitely generated modules. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
finite norm property, relative integral bases, cyclic quartic fields, class number one problem, octic fields, cyclic number fields, normal integral bases, real abelian fields, normal integral basis, quintic fields, sextic fields, relative class number, pure cubic fields, certain algebraic number fields, certain real quadratic fields, biquadratic fields, many imaginary quadratic fields, abelian number fields, conjugate algebraic integers, cubic number fields, having unique factorization, power integral bases, index form equations, relative integral basis, decomposable form equations Key Phrases - Capitalized Phrases (CAPs): (learn more)
Analytical Methods, T-adic Fields, Arithmetical Applications, The Classical Zeta-Functions, Ideal Theorem, Artin's L-functions, Dirichlet's L-functions, Elementary Approach, Distribution of Integers, Riemann Hypothesis, Abelian Galois, Del Corso, Dirichlet L-functions, Extensions Proof, Applying Corollary, Emde Boas, Mac Lane New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
1. This chapter is introductory, and contains the fundamental properties of Dedekind domains including their behaviour under finite extensions and the structure of finitely generated modules. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
finite norm property, relative integral bases, cyclic quartic fields, class number one problem, octic fields, cyclic number fields, normal integral bases, real abelian fields, normal integral basis, quintic fields, sextic fields, relative class number, pure cubic fields, certain algebraic number fields, certain real quadratic fields, biquadratic fields, many imaginary quadratic fields, abelian number fields, conjugate algebraic integers, cubic number fields, having unique factorization, power integral bases, index form equations, relative integral basis, decomposable form equations Key Phrases - Capitalized Phrases (CAPs): (learn more)
Analytical Methods, T-adic Fields, Arithmetical Applications, The Classical Zeta-Functions, Ideal Theorem, Artin's L-functions, Dirichlet's L-functions, Elementary Approach, Distribution of Integers, Riemann Hypothesis, Abelian Galois, Del Corso, Dirichlet L-functions, Extensions Proof, Applying Corollary, Emde Boas, Mac Lane New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 100
books:
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40
books
cite this book:
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- Number Theory by Helmut Hasse on 27 pages
- Number Theory by Wladyslaw Narkiewicz on 8 pages
- Number Theory: An approach through history from Hammurapi to Legendre by André Weil on 6 pages
- Number theory, Volume 20 (Pure and Applied Mathematics) by Author Unknown on 4 pages
- Algebraic Numbers (Pure & Applied Mathematics Monograph) by Paula Ribenboim on page 70, Back Matter (1), and Back Matter (2)
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- Introductory Algebraic Number Theory by Saban Alaca on page 192, and Back Matter
- Factorization in Integral Domains (Lecture Notes in Pure and Applied Mathematics) by Daniel Anderson on page 72, and page 153
- Serre's Problem on Projective Modules by T.Y. Lam in Back Matter
- Algebraic Number Theory by H. Koch in Back Matter
- Lie Algebras and Algebraic Groups (Springer Monographs in Mathematics) by Patrice Tauvel in Back Matter
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