Geodesic
The embedding of a cyclic Galois estension into a cyclic extension
Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 6, Tome 265 (1999), pp. 306-313

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A necessary and sufficient condition of the existence of a cyclic Galois extension of degree $2^n$ containing a given quadratic extension is found. 
@article{ZNSL_1999_265_a22,
     author = {A. V. Yakovlev},
     title = {The embedding of a cyclic {Galois} estension into a cyclic extension},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {306--313},
     publisher = {mathdoc},
     volume = {265},
     year = {1999},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1999_265_a22/}
}
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A. V. Yakovlev. The embedding of a cyclic Galois estension into a cyclic extension. Zapiski Nauchnykh Seminarov POMI, Problems in the theory of representations of algebras and groups. Part 6, Tome 265 (1999), pp. 306-313. http://geodesic.mathdoc.fr/item/ZNSL_1999_265_a22/