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Algebraic Number Theory

J. W. S. Cassels and A. Fröhlich, editors
London Mathematical Society
Publication Date: 
Number of Pages: 
[Reviewed by
Allen Stenger
, on

This is a reprint of a much-admired 1967 text, printed with 8 pages of new errata. The errata were compiled by Kevin Buzzard with the assistance of an army of checkers and are available for download from his web page.

The book is the proceedings of a conference held at Brighton, UK in 1965, but it is much more useful than the typical such collection. This is partly because the conference was organized specifically for teaching, partly because the papers were written by some of the biggest names in the field, and partly because the papers were reworked very carefully by the authors before being published here. The editors included an unusual disclaimer: “neither the lecturers nor the note-takers have any responsibility for any inaccuracies which may remain: they are an act of God.” And indeed, despite the length of the errata, as Buzzard says “almost all of them are utterly trivial.”

The book is still remarkably up-to-date. It covers nearly all areas of the subject, although its approach is slanted somewhat toward class field theory. Some more recent texts with a similar approach and coverage include Lang’s Algebraic Number Theory and Weil’s misnamed Basic Number Theory.

The book’s approach is very abstract and there is very little here on the classic problems that have driven the development of the theory. For example, Fermat’s last theorem, higher-order reciprocity laws, and rational points on elliptic curves are not mentioned except indirectly through some discussion of cyclotomic fields and of complex multiplication. Some more recent texts that give good coverage of these topics include van der Poorten’s Notes on Fermat’s Last Theorem, Ireland & Rosen’s A Classical Introduction to Modern Number Theory, and Silverman & Tate’s Rational Points on Elliptic Curves.

Despite the top-notch contributors and the historical significance of the present book, it is not a good book for beginners in algebraic number theory, primarily because of the lack of applications. It would be quite possible to study the present book and learn a lot from it without learning that the subject had anything to do with numbers. Good books for beginners include Marcus’s Number Fields and Pollard & Diamond’s Theory of Algebraic Numbers, in addition to the application-oriented books listed above. However, these books do not deal with the more advanced topics such as cohomology and class field theory that dominate the present work. This seems like a lot of books to recommend for one subject, but as Lang writes in the Foreword to his book, after recommending a long list of books, “It seems that over the years, everything that has been done has proved useful theoretically or as examples, for the further development of the theory. Old, and seemingly isolated special cases have continuously acquired renewed significance, often after half a century or more.”

Allen Stenger is a math hobbyist and retired software developer. He is webmaster and newsletter editor for the MAA Southwestern Section and is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis. He volunteers in his spare time at, a math help site that fosters inquiry learning.

  • CHAPTER I: Local Fields by A. Fröhlich
    1. Discrete Valuation Rings
    2. Dedekind Domains
    3. Modules and Bilinear Forms
    4. Extensions
    5. Ramification
    6. Totally Ramified Extensions
    7. Non-ramified Extensions
    8. Tamely Ramified Extensions
    9. The Ramification Groups
    10. Decomposition
    • Bibliography
  • CHAPTER II: Global Fields by J. W. S. Cassels
    1. Valuations
    2. Types of Valuation
    3. Examples of Valuations
    4. Topology
    5. Completeness
    6. Independence
    7. Finite Residue Field Case
    8. Normed Spaces
    9. Tensor Product
    10. Extension of Valuations
    11. Extensions of Normalized Valuations
    12. Global Fields
    13. Restricted Topological Product
    14. Adele Ring
    15. Strong Approximation Theorem
    16. Idele Group
    17. Ideals and Divisors
    18. Units
    19. Inclusion and Norm Maps for Adeles, Ideles and Ideals
    • APPENDIX A: Norms and Traces
    • APPENDIX B: Separability
    • APPENDIX C: Hensel's Lemma
  • CHAPTER III: Cyclotomic Fields and Kummer Extensions by B. J. Birch
    1. Cyclotomic Fields
    2. Kummer Extensions
    • APPENDIX: Kummer's Theorem
  • CHAPTER IV: Cohomology of Groups by M. F. Atiyah and C. T. C. Wall (Prepared by I. G. Macdonald on the basis of a manuscript of Atiyah)
    1. Definition of Cohomology
    2. The Standard Complex
    3. Homology
    4. Change of Groups
    5. The Restriction-Inflation Sequence
    6. The Tate Groups
    7. Cup-products
    8. Cyclic Groups: Herbrand Quotient
    9. Cohomological Triviality
    10. Tate's Theorem
  • CHAPTER V: Profinite Groups by K. Gruenberg
    1. The Groups
      1. Introduction
      2. Inverse Systems
      3. Inverse Limits
      4. Topological Characterization of Profinite Groups
      5. Construction of Profinite Groups from Abstract Groups
      6. Profinite Groups in Field Theory
    2. The Cohomology Theory
      1. Introduction
      2. Direct Systems and Direct Limits
      3. Discrete Modules
      4. Cohomology of Profinite Groups
      5. An Example; Generators of pro-p-Groups
      6. Galois Cohomology I: Additive Theory
      7. Galois Cohomology II: "Hilbert 90"
      8. Galois Cohomology III; Brauer Groups
    • References
  • CHAPTER VI: Local Class Field Theory by J-P. Serre (Prepared by J. V. Armitage and J. Neggers)
    • Introduction
    1. The Brauer Group of a Local Field
      1. Statements of Theorems
      2. Computation of H2(Knr/K)
      3. Some Diagrams
      4. Construction of a Subgroup with Trivial Cohomology
      5. An Ugly Lemma
      6. End of Proofs
      7. An Auxiliary Result
      • APPENDIX: Division Algebras Over a Local Field
    2. Abelian Extensions of Local Fields
      1. Cohomological Properties
      2. The Reciprocity Map
      3. Characterization of (α, L/K) by Characters
      4. Variations with the Fields Involved
      5. Unramified Extensions
      6. Norm Subgroups
      7. Statement of the Existence Theorem
      8. Some Characterizations of (α, L/K)
      9. The Archimedean Case
    3. Formal Multiplication in Local Fields
      1. The Case K = Qp
      2. Formal Groups
      3. Lubin-Tate Formal Group Laws
      4. Statements
      5. Construction of Ff and [a]f
      6. First Properties of the Extension Kπ of K
      7. The Reciprocity Map
      8. The Existence Theorem
    4. Ramification Subgroups and Conductors
      1. Ramification Groups
      2. Abelian Conductors
      3. Artin's Conductors
      4. Global Conductors
      5. Artin's Representation
  • CHAPTER VII: Global Class Field Theory by J. T. Tate (Prepared by B. J. Birch and R. R. Laxton)
    1. Action of the Galois Group on Primes and Completions
    2. Frobenius Automorphisms
    3. Artin's Reciprocity Law
    4. Chevalley's Interpretation by Idèles
    5. Statement of the Main Theorems on Abelian Extensions
    6. Relation between Global and Local Artin Maps
    7. Cohomology of Idèles
    8. Cohomology of Idèle Classes (I): The First Inequality
    9. Cohomology of Idèle Classes (II): The Second Inequality
    10. Proof of the Reciprocity Law
    11. Cohomology of Idèle Classes (III): The Fundamental Class
    12. Proof of the Existence Theorem
    • List of Symbols
  • CHAPTER VIII: Zeta-Functions and L-Functions by H. Heilbronn (Prepared by D. A. Burgess and H. Halberstam)
    1. Characters
    2. Dirichlet L-series and Density Theorems
    3. L-functions for Non-abelian Extensions
    • References
  • CHAPTER IX: On Class Field Towers by Peter Roquette
    1. Introduction
    2. Proof of Theorem 3
    3. Proof of Theorem 5 for Galois Extensions
    • References
  • CHAPTER X: Semi-Simple Algebraic Groups by M. Kneser (Prepared by I. G. Macdonald)
    • Introduction
    1. Algebraic Theory
      1. Algebraic Groups over an Algebraically Closed Field
      2. Semi-Simple Groups over an Algebraically Closed Field
      3. Semi-Simple Groups over Perfect Fields
    2. Galois Cohomology
      1. Non-Commutative Cohomology
      2. K-forms
      3. Fields of Dimension ≤ 1
      4. p-adic Fields
      5. Number Fields
    3. Tamagawa Numbers
      • Introduction
      1. The Tamagawa Measure
      2. The Tamagawa Number
      3. The Theorem of Minkowski and Siegel
    • References
  • CHAPTER XI: History of Class Field Theory by Helmut Hasse (Prepared by A. Lue on the basis of a preliminary manuscript of the lecturer)
    • References
  • CHAPTER XII: An Application of Computing to Class Field Theory by H. P. F. Swinnerton-Dyer
  • CHAPTER XIII: Complex Multiplication by J-P. Serre (Prepared by B. J. Birch)
    • Introduction
    1. The Theorems
    2. The Proofs
    3. Maximal Abelian Extension
    • References
  • CHAPTER XIV: l-Extensions by K. Hoechsmann
    • Introduction
    1. Two Lemmas
    2. Local Fields
    3. Global Fields
    • APPENDIX: Restricted Ramification
    • References
  • CHAPTER XV: Fourier Analysis in Number Fields and Hecke's Zeta-Functions by J. T. Tate (Thesis, 1950)
    1. Introduction
      1. Relevant History
      2. This Thesis
      3. "Prerequisites"
    2. The Local Theory
      1. Introduction
      2. Additive Characters and Measure
      3. Multiplicative Character and Measure
      4. The Local ζ-Functions; Functional Equation
      5. Computation of ρ(c) by Special ζ-Functions
    3. Abstract Restricted Direct Product
      1. Introduction
      2. Characters
      3. Measure
    4. The Theory in the Large
      1. Additive Theory
      2. Riemann-Roch Theorem
      3. Multiplicative Theory
      4. The ζ-Functions; Functional Equation
      5. Comparison with the Classical Theory
    • A few Comments on Recent Related Literature
    • References
  • EXERCISES (prepared by Tate and Serre)
    • Exercise 1: The Power Residue Symbol
    • Exercise 2: The Norm Residue Symbol
    • Exercise 3: The Hilbert Class Field
    • Exercise 4: Numbers Represented by Quadratic Forms
    • Exercise 5: Local Norms not Global Norms
    • Exercise 6: On Decomposition of Primes
    • Exercise 7: A Lemma on Admissible Maps
    • Exercise 8: Norms from Non-Abelian Extensions