Abstract
AbstractLet $$Q_8$$
Q
8
be the quaternion group of order 8 and $${\chi }$$
χ
its faithful irreducible character. Then $${\chi }$$
χ
can be realized over certain imaginary quadratic number fields $$K={\mathbb Q}\bigl (\sqrt{-N}\bigr )$$
K
=
Q
(
-
N
)
but not over their rings of integers (Feit, Serre); here N is a positive square-free integer. We show that this happens precisely when $${\mathbb Q}\bigl (\sqrt{N}\bigr )$$
Q
(
N
)
but not $${\mathbb Q}\bigl (\sqrt{2}, \sqrt{N}\bigr )$$
Q
(
2
,
N
)
can be embedded into a $$Q_8$$
Q
8
-field over the rationals (Galois with group $$Q_8$$
Q
8
) and N is not a sum of two integer squares. In particular, we get that $${\chi }$$
χ
cannot be integrally realized if N is (properly) divisible by some prime $$q\equiv 7\,({\textrm{mod}\,}8)$$
q
≡
7
(
mod
8
)
.
Funder
Eberhard Karls Universität Tübingen
Publisher
Springer Science and Business Media LLC
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