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

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 Z 2 \mathbb Z_2 -extension of the rationals. The list of such sets yields new affirmative examples of Greenberg’s conjecture.