Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

For a finite group $G$ let ${\cal K}_2(G)$ denote the set of normal number fields (within ${\mathbb C}$) with Galois group $G$ which are $2$-ramified, that is, unramified outside $\{2,\infty \}$. We describe the $2$-groups $G$ for which ${\cal K}_2(G)\not =\varnothing $, and determine the fields in ${\cal K}_2(G)$ 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).