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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Unramified class field theory for orders
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by Peter Stevenhagen PDF
Trans. Amer. Math. Soc. 311 (1989), 483-500 Request permission


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.
  • M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802
  • Hyman Bass, Algebraic $K$-theory, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0249491
  • N. Bourbaki, Commutative algebra, Addison-Wesley, Reading, Mass., 1972.
  • Charles W. Curtis and Irving Reiner, Methods of representation theory. Vol. I, Pure and Applied Mathematics, John Wiley & Sons, Inc., New York, 1981. With applications to finite groups and orders. MR 632548
  • RevĂȘtements Ă©tales et groupe fondamental, Lecture Notes in Mathematics, Vol. 224, Springer-Verlag, Berlin-New York, 1971 (French). SĂ©minaire de GĂ©omĂ©trie AlgĂ©brique du Bois Marie 1960–1961 (SGA 1); DirigĂ© par Alexandre Grothendieck. AugmentĂ© de deux exposĂ©s de M. Raynaud. MR 0354651
  • Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
  • Philip J. Higgins, Notes on categories and groupoids, Van Nostrand Rienhold Mathematical Studies, No. 32, Van Nostrand Reinhold Co., London-New York-Melbourne, 1971. MR 0327946
  • David Hilbert, Über die Theorie der relativ-Abel’schen Zahlkörper, Acta Math. 26 (1902), no. 1, 99–131 (German). MR 1554953, DOI 10.1007/BF02415486
  • Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
  • H. W. Lenstra, Jr., Galois theory for schemes, Mathematisch Instituut, Universiteit van Amsterdam, 1985.
  • James S. Milne, Étale cohomology, Princeton Mathematical Series, No. 33, Princeton University Press, Princeton, N.J., 1980. MR 559531
  • John Milnor, Introduction to algebraic $K$-theory, Annals of Mathematics Studies, No. 72, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1971. MR 0349811
  • J. P. Murre, Lectures on an introduction to Grothendieck’s theory of the fundamental group, Tata Institute of Fundamental Research Lectures on Mathematics, No. 40, Tata Institute of Fundamental Research, Bombay, 1967. Notes by S. Anantharaman. MR 0302650
  • P. Stevenhagen, Generalized unramified class field theory, Mathematisch Instituut, Universiteit van Amsterdam, report 85-13, 1985.
  • Edwin Weiss, Algebraic number theory, McGraw-Hill Book Co., Inc., New York-San Francisco-Toronto-London, 1963. MR 0159805
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Additional Information
  • © Copyright 1989 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 311 (1989), 483-500
  • MSC: Primary 11R37; Secondary 11R54
  • DOI:
  • MathSciNet review: 978366