Quaternion extensions with restricted ramification

Peter Schmid. "Quaternion extensions with restricted ramification." Acta Arithmetica 165.2 (2014): 123-140. <http://eudml.org/doc/279114>.

@article{PeterSchmid2014,
abstract = {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 $p ≡ 1 (mod 2^\{n-1\})$, 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 $Q_\{2^n\}$ of order $2^n$.},
author = {Peter Schmid},
journal = {Acta Arithmetica},
keywords = {Galois theory; quaternion extensions; restricted ramification},
language = {eng},
number = {2},
pages = {123-140},
title = {Quaternion extensions with restricted ramification},
url = {http://eudml.org/doc/279114},
volume = {165},
year = {2014},
}

TY - JOUR
AU - Peter Schmid
TI - Quaternion extensions with restricted ramification
JO - Acta Arithmetica
PY - 2014
VL - 165
IS - 2
SP - 123
EP - 140
AB - 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 $p ≡ 1 (mod 2^{n-1})$, 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 $Q_{2^n}$ of order $2^n$.
LA - eng
KW - Galois theory; quaternion extensions; restricted ramification
UR - http://eudml.org/doc/279114
ER -