In any normal number field having Q₈, 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 Q₈-fields over the rationals where only two (finite) primes are ramified. We also show that, for any integer n>3 and any prime , there exist unique real and complex normal number fields which are unramified outside S = 2,p and cyclic over ℚ(√2) and whose Galois group is the (generalized) quaternion group of order .
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-2-2,
author = {Peter Schmid},
title = {Quaternion extensions with restricted ramification},
journal = {Acta Arithmetica},
volume = {166},
year = {2014},
pages = {123-140},
zbl = {1317.11112},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-2-2}
}
Peter Schmid. Quaternion extensions with restricted ramification. Acta Arithmetica, Tome 166 (2014) pp. 123-140. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa165-2-2/