Embedding of a cyclic extension into a cyclic extension (Q1866701)

Let \(k\) be a field, \(K/k\) be cyclic of order \(m\), and let \(n\) be a multiple of \(m\). The extension \(K/k\) is called \(n\)-embeddable if there exists a cyclic extension \(L/k\) such that \([L:k]=n\) and \(K\subseteq L\). The author gives necessary and sufficient conditions for \(K/k\) to be \(2^s\)-embeddable in the case \([K:k]=2\), \(\text{char}(k)\neq 2\). The problem is a particular case of the Galois embedding problem with cyclic kernel of even order solved completely by \textit{P. Schmidt} and the author [Vestn. Leningr. Univ. 18, No. 13, Ser. Mat. Mekh. Astron. No. 3, 137-139 (1963; Zbl 0126.27403)], but it is not easy to deduce the conditions given here from the general result.