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Abstract

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 $2^n$.