Quaternion lattices and quaternion fields

Springer Science and Business Media LLC - 2023
Peter Schmid1
1Mathematisches Institut der Universität Tübingen, Tübingen, Germany

Tóm tắt

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

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