Tame pro-2 Galois groups and the basic ℤ₂-extension

Mizusawa, Yasushi

doi:10.1090/tran/7023

Tame pro-2 Galois groups and the basic $\mathbb {Z}_2$-extension
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Trans. Amer. Math. Soc. 370 (2018), 2423-2461 Request permission

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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