Tame pro-2 Galois groups and the basic $\mathbb {Z}_2$-extension
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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.References
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Additional Information
- Yasushi Mizusawa
- Affiliation: Department of Mathematics, Nagoya Institute of Technology, Gokiso, Showa, Nagoya 466-8555, Japan
- MR Author ID: 672607
- Email: mizusawa.yasushi@nitech.ac.jp
- 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)
- © Copyright 2017 American Mathematical Society
- 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
- MathSciNet review: 3748573