Quaternion extensions with restricted ramification

Peter Schmid¹

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

Acta Arithmetica, Tome 165 (2014) no. 2, pp. 123-140

Résumé

In any normal number field having $Q_8$, the quaternion group of order $8$, as Galois group over the rationals, at least two finite primes must ramify. The classical example by Dedekind of such a field is extraordinary in that it is totally real and only the primes $2$ and $3$ are ramified. In this note we describe in detail all $Q_8$-fields over the rationals where only two (finite) primes are ramified. We also show that, for any integer $n>3$ and any prime $p\equiv 1\ ({\rm mod}\ 2^{n-1})$, there exist unique real and complex normal number fields which are unramified outside $S=\{2,p\}$ and cyclic over ${\mathbb Q}(\sqrt 2)$ and whose Galois group is the (generalized) quaternion group $Q_{2^n}$ of order $2^n$.

Détail

DOI : 10.4064/aa165-2-2

Keywords: normal number field having quaternion group order galois group rationals least finite primes ramify classical example dedekind field extraordinary totally real only primes ramified note describe detail fields rationals where only finite primes ramified integer prime equiv mod n there exist unique real complex normal number fields which unramified outside cyclic mathbb sqrt whose galois group generalized quaternion group order