For a finite group G let 𝒦₂(G) denote the set of normal number fields (within ℂ) with Galois group G which are 2-ramified, that is, unramified outside {2,∞}. We describe the 2-groups G for which 𝒦₂(G) ≠ ∅, and determine the fields in 𝒦₂(G) for certain distinguished 2-groups G appearing (dihedral, semidihedral, modular and semimodular groups). Our approach is based on Fröhlich's theory of central field extensions, and makes use of ring class field constructions (complex multiplication).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-2-2,
author = {Peter Schmid},
title = {On 2-extensions of the rationals with restricted ramification},
journal = {Acta Arithmetica},
volume = {166},
year = {2014},
pages = {111-125},
zbl = {1305.11099},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-2-2}
}
Peter Schmid. On 2-extensions of the rationals with restricted ramification. Acta Arithmetica, Tome 166 (2014) pp. 111-125. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa163-2-2/