Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Unramified class field theory for orders

Author: Peter Stevenhagen
Journal: Trans. Amer. Math. Soc. 311 (1989), 483-500
MSC: Primary 11R37; Secondary 11R54
MathSciNet review: 978366
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The main theorem of unramified class field theory, which states that the class group of the ring of integers of a number field $ K$, is canonically isomorphic to the Galois group of the maximal totally unramified abelian extension of $ K$ over $ K$, is generalized and proved for all infinite commutative rings with unit that, like rings of integers, are connected and finitely generated as a module over $ {\mathbf{Z}}$. Modulo their nilradical, these rings are exactly the connected orders in products of number fields.

References [Enhancements On Off] (What's this?)

  • [1] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, Mass., 1969. MR 0242802 (39:4129)
  • [2] H. Bass, Algebraic $ K$-theory, Benjamin, New York, 1968. MR 0249491 (40:2736)
  • [3] N. Bourbaki, Commutative algebra, Addison-Wesley, Reading, Mass., 1972.
  • [4] C. W. Curtis and I. Reiner, Methods of representation theory, Vol. I, Wiley, New York, 1981. MR 632548 (82i:20001)
  • [5] A. Grothendieck, Séminaire de géométrie algébrique $ 1$, Revêtements étales et groupe fondamental, Lecture Notes in Math., vol. 224, Springer, Berlin, 1971. MR 0354651 (50:7129)
  • [6] R. Hartshorne, Algebraic geometry, Springer, New York, 1977. MR 0463157 (57:3116)
  • [7] P. J. Higgins, Notes on categories and groupoids, Van Nostrand Reinhold, London, 1971. MR 0327946 (48:6288)
  • [8] D. Hilbert, Über die Theorie der relativ-Abelschen Zahlkörper, Acta Math. 26 (1902), 99-132. [Only this `mit geringen Änderungen' reprinted version of a paper under the same title in the Nachr. d. Ges. der Wiss. zu Gött. (1898), 370-399, contains the paragraph on the Hilbert class field.] MR 1554953
  • [9] S. Lang, Algebraic number theory, Addison-Wesley, Reading, Mass., 1970. MR 0282947 (44:181)
  • [10] H. W. Lenstra, Jr., Galois theory for schemes, Mathematisch Instituut, Universiteit van Amsterdam, 1985.
  • [11] J. S. Milne, Etale cohomology, Princeton Univ. Press, Princeton, N.J., 1980. MR 559531 (81j:14002)
  • [12] J. Milnor, Introduction to algebraic $ K$-theory, Princeton Univ. Press, Princeton, N.J., 1971. MR 0349811 (50:2304)
  • [13] J. P. Murre, Lectures on an introduction to Grothendieck's theory of the fundamental group, Tata Institute of Fundamental Research, Bombay, 1967. MR 0302650 (46:1794)
  • [14] P. Stevenhagen, Generalized unramified class field theory, Mathematisch Instituut, Universiteit van Amsterdam, report 85-13, 1985.
  • [15] E. Weiss, Algebraic number theory, McGraw-Hill, New York, 1963. MR 0159805 (28:3021)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 11R37, 11R54

Retrieve articles in all journals with MSC: 11R37, 11R54

Additional Information

Keywords: Artin isomorphism, Grothendieck group, algebraic fundamental group, van Kampen theorem
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society