On 2-extensions of the rationals with restricted ramification

Peter Schmid¹

¹ Mathematisches Institut Universität Tübingen Auf der Morgenstelle 10 72076 Tübingen, Germany

Acta Arithmetica, Tome 163 (2014) no. 2, pp. 111-125

Résumé

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).

Détail

DOI : 10.4064/aa163-2-2

Keywords: finite group cal denote set normal number fields within mathbb galois group which ramified unramified outside infty describe groups which cal varnothing determine fields cal certain distinguished groups appearing dihedral semidihedral modular semimodular groups approach based hlichs theory central field extensions makes ring class field constructions complex multiplication