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Transactions of the American Mathematical Society

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Unramified class field theory for orders


Author: Peter Stevenhagen
Journal: Trans. Amer. Math. Soc. 311 (1989), 483-500
MSC: Primary 11R37; Secondary 11R54
DOI: https://doi.org/10.1090/S0002-9947-1989-0978366-3
MathSciNet review: 978366
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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.


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Additional Information

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