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Transactions of the American Mathematical Society

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



An elementary proof of the local Kronecker-Weber theorem

Author: Michael Rosen
Journal: Trans. Amer. Math. Soc. 265 (1981), 599-605
MSC: Primary 12B15
MathSciNet review: 610968
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Abstract: Let $ K$ be a local field. Lubin and Tate have shown how to explicitly construct an abelian extension of $ K$ which they prove to be the maximal abelian extension. Their proof of this result uses local class field theory. When $ K$ is a $ p$-adic field we give an elementary proof which even avoids the use of higher ramification groups. Instead we rely on facts about the principal units in a finite abelian extension of $ K$ as a module for the Galois group.

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Keywords: Local field, abelian extension, Kummer theory, principal units, group ring, Lubin-Tate construction
Article copyright: © Copyright 1981 American Mathematical Society

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