Open AccessTame pro-2 Galois groups and the basic ℤ₂-extension
Author(s) -
Yasushi Mizusawa
Publication year - 2017
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/tran/7023
Subject(s) - mathematics , galois group , prime (order theory) , conjecture , galois module , extension (predicate logic) , galois extension , galois theory , discrete mathematics , combinatorics , computer science , programming language
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.