Some field embedding problem with cyclic kernel

The problem of embedding a quadratic extension of a number field into an extension with a cyclic 2-group is studied. A reduction theorem showing that, under the compatibility condition, an additional condition of embedding consists of the solvability of the problem with cyclic kernel of order 16 (of course, the degree of the desired field is no less than 16.) An additional condition of the embedding into a field of degree 16 is found; namely, the number generating the given quadratic extension must be a norm in the cyclotomic field containing the primitive roots of unity of eight degree. For $Q$, the condition of embedding is easier: all odd prime divisors of the generating element must be congruent with 1 modulo the order of the extension group. In addition, the quadratic extension must be real.