Book Review
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Book Information:
Authors:
Jürgen Neukirch,
Alexander Schmidt and
Kay Wingberg
Title:
Cohomology of number fields
Additional book information:
Grundlehren der mathematischen Wissenschaften, vol. 323,
Springer-Verlag,
2000,
720 pp.,
ISBN 3-540-66671-0,
$109.00$
1. E. Artin and J. Tate, Class field theory, second ed., Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1990. Originally published by W. A. Benjamin in 1968. MR 1043169; MR 36:6383
2. J. W. S. Cassels and A. Fröhlich (eds.), Algebraic number theory (Brighton, 1965), Academic Press, 1967. MR 0215665
3. C. Chevalley, La théorie du corps de classes, Ann. of Math. 41 (1940), 394-418. MR 2:38c
4. H. Hasse, History of class field theory, In Cassels and Fröhlich [2], pp. 266-279. MR 0218330
5. D. Hilbert, The theory of algebraic number fields, Springer-Verlag, 1998. MR 1646901
6. G. Hochschild, Local class field theory, Ann. of Math. 51 (1950), 331-347. MR 11:490a
7. -, Note on Artin's reciprocity law, Ann. of Math. 52 (1950), 694-701. MR 12:315c
8. G. Hochschild and T. Nakayama, Cohomology in class field theory, Ann. of Math. 55 (1952), 348-366. MR 13:916d
9. H. Hopf, Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv. 14 (1942), 257-309. MR 3:316e
10. W. Hurewicz, Beiträge zur Topologie der Deformationen, Proc. Akad. Amsterdam 38 (1936), 112-119, 521-538, and 39 (1936), 117-125, 215-224.
11. S. MacLane, Origins of the cohomology of groups, Enseign. Math. 24 (1978), 1-29. MR 0497280
12. J. S. Milne, Arithmetic duality theorems, Academic Press, 1986. MR 0881804
13. T. Nakayama, Idèle-class factor sets and class field theory, Ann. of Math. 55 (1952), 73-84. MR 13:629a
14. -, On a 3-cohomology class in class field theory and the relationship of algebra- and idèle-classes, Ann. of Math. 57 (1953), 1-14. MR 14:453a
15. J. Neukirch, Class field theory, Springer-Verlag, 1986. MR 0819231
16. -, Algebraic number theory, Springer-Verlag, 1999. Translated from the 1992 German edition by N. Schappacher. MR 1697859
17. J.-P. Serre, Local fields, Springer-Verlag, 1979. MR 0554237
18. J.-P. Serre, Galois cohomology, Springer-Verlag, Berlin, 1997. Translated from the French by Patrick Ion and revised by the author. MR 1466966
19. S. Takahashi, Homology groups in class field theory, Tôhoku Math. J. 5 (1953), 8-11. MR 15:606b
20. J. Tate, The higher dimensional cohomology groups of class field theory, Ann. of Math. 56 (1952), 294-297. MR 14:252b
21. A. Weil, Sur la théorie du corps de classes, J. Math. Soc. Japan 3 (1951), 1-35. MR 13:439d
22. -, Basic number theory, Springer-Verlag, 1967. MR 0234930
- 1.
- E. Artin and J. Tate, Class field theory, second ed., Addison-Wesley Publishing Company Advanced Book Program, Redwood City, CA, 1990. Originally published by W. A. Benjamin in 1968. MR 1043169; MR 36:6383
- 2.
- J. W. S. Cassels and A. Fröhlich (eds.), Algebraic number theory (Brighton, 1965), Academic Press, 1967. MR 0215665
- 3.
- C. Chevalley, La théorie du corps de classes, Ann. of Math. 41 (1940), 394-418. MR 2:38c
- 4.
- H. Hasse, History of class field theory, In Cassels and Fröhlich [2], pp. 266-279. MR 0218330
- 5.
- D. Hilbert, The theory of algebraic number fields, Springer-Verlag, 1998. MR 1646901
- 6.
- G. Hochschild, Local class field theory, Ann. of Math. 51 (1950), 331-347. MR 11:490a
- 7.
- -, Note on Artin's reciprocity law, Ann. of Math. 52 (1950), 694-701. MR 12:315c
- 8.
- G. Hochschild and T. Nakayama, Cohomology in class field theory, Ann. of Math. 55 (1952), 348-366. MR 13:916d
- 9.
- H. Hopf, Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv. 14 (1942), 257-309. MR 3:316e
- 10.
- W. Hurewicz, Beiträge zur Topologie der Deformationen, Proc. Akad. Amsterdam 38 (1936), 112-119, 521-538, and 39 (1936), 117-125, 215-224.
- 11.
- S. MacLane, Origins of the cohomology of groups, Enseign. Math. 24 (1978), 1-29. MR 0497280
- 12.
- J. S. Milne, Arithmetic duality theorems, Academic Press, 1986. MR 0881804
- 13.
- T. Nakayama, Idèle-class factor sets and class field theory, Ann. of Math. 55 (1952), 73-84. MR 13:629a
- 14.
- -, On a 3-cohomology class in class field theory and the relationship of algebra- and idèle-classes, Ann. of Math. 57 (1953), 1-14. MR 14:453a
- 15.
- J. Neukirch, Class field theory, Springer-Verlag, 1986. MR 0819231
- 16.
- -, Algebraic number theory, Springer-Verlag, 1999. Translated from the 1992 German edition by N. Schappacher. MR 1697859
- 17.
- J.-P. Serre, Local fields, Springer-Verlag, 1979. MR 0554237
- 18.
- J.-P. Serre, Galois cohomology, Springer-Verlag, Berlin, 1997. Translated from the French by Patrick Ion and revised by the author. MR 1466966
- 19.
- S. Takahashi, Homology groups in class field theory, Tôhoku Math. J. 5 (1953), 8-11. MR 15:606b
- 20.
- J. Tate, The higher dimensional cohomology groups of class field theory, Ann. of Math. 56 (1952), 294-297. MR 14:252b
- 21.
- A. Weil, Sur la théorie du corps de classes, J. Math. Soc. Japan 3 (1951), 1-35. MR 13:439d
- 22.
- -, Basic number theory, Springer-Verlag, 1967. MR 0234930
Review Information:
Reviewer:
Fernando Q. Gouvêa
Affiliation:
Colby College
Email:
fqgouvea@colby.edu
Journal:
Bull. Amer. Math. Soc.
39 (2002), 101-107
Published electronically:
October 10, 2001
Review copyright:
© Copyright 2001
American Mathematical Society
