An elementary proof of the local Kronecker-Weber theorem
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- by Michael Rosen PDF
- Trans. Amer. Math. Soc. 265 (1981), 599-605 Request permission
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.References
- Z. I. Borevič, The multiplicative group of cyclic $p$-extensions of a local field, Trudy Mat. Inst. Steklov 80 (1965), 16–29 (Russian). MR 0205976
- Michiel Hazewinkel, Local class field theory is easy, Advances in Math. 18 (1975), no. 2, 148–181. MR 389858, DOI 10.1016/0001-8708(75)90156-5 M. Krasner, Sur la representation exponentielle dans les corp relativement galoisien de nombres $p$-adique, Acta Arith. 3 (1939), 133-173. S. Lang, Algebraic number theory, Addison-Wesley, Reading, Mass., 1968.
- Jonathan Lubin and John Tate, Formal complex multiplication in local fields, Ann. of Math. (2) 81 (1965), 380–387. MR 172878, DOI 10.2307/1970622
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 265 (1981), 599-605
- MSC: Primary 12B15
- DOI: https://doi.org/10.1090/S0002-9947-1981-0610968-9
- MathSciNet review: 610968