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Given a quadratic number field $F = \mathbb{Q}(\sqrt{d})$, is there a way to determine whether or not $F \subset K$ for some quartic numberfield $K$ with $\operatorname{Gal}(K/\mathbb{Q}) \cong \mathbb{Z}/4\mathbb{Z}$?

Moreover, how can we find that field $K$ if it exists?

For instance, $\mathbb{Q}(i)$ is not contained in any cyclic quartic number field and neither is $\mathbb{Q}(\sqrt{-3})$. I thought I heard somewhere that the intermediate subfield of a cyclic quartic number field must be totally real, but I am not sure.

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  • That sounds reasonable - if not $K$ is not totally real, then Gal($L/K$) must contain complex conjugation, which is of order 2. – Mathmo123 Jun 24 '14 at 23:55
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    Note that $Aut(F/\mathbb{Q}) \subset Aut(K/\mathbb{Q})$. If $F$ were totally real, then there would be two automorphisms of order $2$. Namely, $\phi \in Aut(F/\mathbb{Q})$ such that $\phi(\sqrt{d}) = -\sqrt{d}$, and another from complex conjugation. However, $\mathbb{Z}_4$ has only one element of order $2$. The moral of the story is that $\sqrt{d} \in \mathbb{C}$ \ $\mathbb{R}$ if this is to work. – Kaj Hansen Jun 25 '14 at 00:58

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