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Transactions of the American Mathematical Society

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Tame pro-2 Galois groups and the basic $ \mathbb{Z}_2$-extension


Author: Yasushi Mizusawa
Journal: Trans. Amer. Math. Soc. 370 (2018), 2423-2461
MSC (2010): Primary 11R23; Secondary 11R18, 11R20, 11R32
DOI: https://doi.org/10.1090/tran/7023
Published electronically: October 31, 2017
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Abstract: For a number field, we consider the Galois group of the maximal tamely ramified pro-2-extension with restricted ramification. Providing a general criterion for the metacyclicity of such Galois groups in terms of 2-ranks and 4-ranks of ray class groups, we classify all finite sets of odd prime numbers such that the maximal pro-2-extension unramified outside the set has prometacyclic Galois group over the $ \mathbb{Z}_2$-extension of the rationals. The list of such sets yields new affirmative examples of Greenberg's conjecture.


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Yasushi Mizusawa
Affiliation: Department of Mathematics, Nagoya Institute of Technology, Gokiso, Showa, Nagoya 466-8555, Japan
Email: mizusawa.yasushi@nitech.ac.jp

DOI: https://doi.org/10.1090/tran/7023
Received by editor(s): April 30, 2016
Received by editor(s) in revised form: July 14, 2016
Published electronically: October 31, 2017
Additional Notes: This work was supported by JSPS KAKENHI Grant Number JP26800010, Grant-in-Aid for Young Scientists (B)
Article copyright: © Copyright 2017 American Mathematical Society

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