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Unramified quaternion extensions of quadratic number fields. (English) Zbl 0890.11031

This paper is concerned with the construction of unramified quaternion extensions of number fields, particularly of quadratic number fields. (Quaternion extensions of the rationals always ramify over their biquadratic subfield.) For a quadratic number field, the existence of an unramified quaternion extension which is normal over the rationals is shown to be equivalent to an elementary condition on the factorization of its discriminant. It is further shown explicitly how to construct such extensions using elementary methods. An analogous result is stated for unramified dihedral extensions of quadratic number fields.

MSC:

11R21 Other number fields
11R11 Quadratic extensions
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