Abstract
Let K be any number field, and let E be a quadratic or a biquadratic extension of K. We give necessary conditions for the existence of a field N, respectively, cyclic of degree 4, dihedral or quaternionian of degree 8, over K, which admits a normal integral basis over E. When the class number of K is h( K) = 1, we prove that these conditions are also sufficient, and we provide explicit formulae to construct normal integral bases of N over E.
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