Embedding quadratic fields into cyclic extension Let $K$ be a quadratic number field, I know that $K$ can be embedded into a cyclic extension $L/\Bbb Q$ of degree $4$ if and only if $-1 \in N_{K/\Bbb Q}(K)$. What about the higher degree case?
In other words, for every quadratic number field $K$ and every $n \in \Bbb N$ we can consider the following question $(n)$:
Could $K$ be embedded into a cyclic extension $L/\Bbb Q$ of degree $2^n$?
In particular, does there exist a $n_0 \in \Bbb N$ such that if question $(n_0)$ is true for some $K$ then question $(n)$ is true for this $K$ for every $n$ ?
 A: Let us fix beforehand a convenient vocabulary. Given a prime $p$, a cyclic Galois field extension $L/k$ of degree $p^n$ is called a $\mathbf Z/p^n$-extension. For $m\ge n$, we say that $L/k$ is $\mathbf Z/p^m$- embeddable if it can be embedded into a $\mathbf Z/p^m$-extension $L'/k$. Then the answer to your question - which is not at all elementary - will be three-fold:
1) Let $L/k$ be a $\mathbf Z/p^n$-extension such that that $k$ contains a primitive $p^r$-th root of unity $\zeta$. Then $L/k$ is $\mathbf Z/p^{n+r}$- embeddable iff $\zeta$ is a norm from $L$. This is a purely  Galois theoretical exercise, see e.g. Artin-Tate, "Class Field Theory", chap.10, coroll.2. In your case, $k=\mathbf Q, \zeta = -1$ and the above just says that a quadratic field $K$ is $\mathbf Z/4$- embeddable iff $-1$ is a norm from $K$. A criterion for $\mathbf Z/2^m$- embeddability with $m\ge 3$ obviously needs additional ingredients.
2) From now on, a $\mathbf Z/p^n$-extension of number fields $L/k$ will be called $p^{\infty}$ - embeddable if it is $\mathbf Z/p^m$- embeddable for all $m\ge n$, and $\mathbf Z_p$- embeddable if it can be embedded into a $\mathbf Z_p$-extension of $k$, i.e. an infinite Galois extension $ k_\infty /k$ with Galois group isomorphic to the additive group $\mathbf Z_p$ of the $p$-adic integers. Obviously $\mathbf Z_p$- embeddability implies infinite embeddability, but the two properties are not equivalent in general. Let $S$ be the set of places of $k$ above $p$. The key ingredient  will be the notion of $S$-ramification, i.e. of extensions unramified at every place outside $S$. Let $\tilde k$ (resp. $\hat k$) be the compositum of all the $\mathbf Z_p$-extensions (resp. all the $p^{\infty}$ - embeddable $\mathbf Z/p^n$ - extensions of $k$, for all $n$), and $k_S$ the maximal $S$-ramified abelian pro-$p$-extension of $k$. It follows from  class field theory that $\tilde k \subset \hat k \subset k_S $ (see e.g Bertrandias & Payan, "$\Gamma$-extensions et invariants cyclotomiques", Ann. Sci. ENS, 1972, 517-543).
3) For a general number field $k$, the determination of $Gal(k_S/k)$ is quite a difficult problem: the $ \mathbf Z_p$-rank is given by the famous Leopoldt conjecture, and the $\mathbf Z_p$-torsion is related to the $p$-adic L-functions by the so called Main Conjecture of Iwasawa theory. Fortunately, for $k=\mathbf Q$, it is known from CFT that the rank is $1$ and the torsion is $0$, i.e. $Gal(\mathbf Q_S/\mathbf Q) =\mathbf Z_p$ . It follows that the only $p^{\infty}$ - embeddable $\mathbf Z/p^n$ -extensions of $\mathbf Q$ are the  $\mathbf Z/p^n$ -subextensions of the unique $\mathbf Z_p$-extension  of $\mathbf Q$. In particular, a quadratic field $K$ is $\mathbf Z/2^n$- embeddable  for all $n$ iff $K$ is the quadratic subfield of the unique $\mathbf Z_2$-extension  of $\mathbf Q$ ./.
