Let $K/\mathbb{Q}$ be a degree $4$ number field. Is it known how to determine whether the Galois closure of $K$, say $K'$, contains the 4-th roots of unity?


In the cubic case, there is a succinct answer: the Galois closure $E'$ of a cubic field $E/\mathbb{Q}$ contains the third roots of unity if and only if $E$ is a pure cubic field: that is, $E = \mathbb{Q}(\sqrt[3]{n})$ for some cube-free integer $n$. 


Such a simple characterization doesn't seem to work for the quartic case, since for example the field $\mathbb{Q}(\sqrt{a}, \sqrt{-1})$ with a square-free integer $a \ne \pm 1$ contains the $4$-th roots of unity but is not of the form $\mathbb{Q}(\sqrt[4]{n})$ for a $4$-free integer $n$. 


If this characterization is known classically, can someone give a reference? 



Which quartic fields contain the 4th roots of unity in their Galois closure?
As explained in the comments, the only non-trivial case is where $K/\mathbb{Q}$ has Galois closure $K'/\mathbb{Q}$ with $\operatorname{Gal}(K'/\mathbb{Q})$ isomorphic to $D\_4$. I will do this case here since it is too long for a comment.


In that case, since all proper subgroups of $D\_4$ are contained in a subgroup of index $2$, we must have a quadratic subextension $F/\mathbb{Q}$, so write $F = \mathbb{Q}(\sqrt{d})$. Also write $K = F(\sqrt{\delta})$, with $\delta = a + b \sqrt{d}$, where $a,b$ are in $\mathbb{Q}$.


Since $D\_4$ has three subgroups of index $2$, $K'$ has three quadratic subextensions, of which only one can be contained in $K$, since otherwise we would have had the Klein $4$-group as our Galois group. Now $K'$ is generated over $K$ by an element $\sqrt{\delta'}$ where $\delta' = a - b \sqrt{d}$.



> 
> **Claim**. We have that $\operatorname{Nm}\_{F/\mathbb{Q}}(\delta)=\delta\delta'=a^2-db^2$ is not contained in $F^{\times 2} \cap \mathbb{Q}^{\times}$, which by Kummer theory is the subgroup of $\mathbb{Q}^{\times}$ generated by $\mathbb{Q}^{\times 2}$ and $d$. 
> 



> 
> *Proof.* This says precisely that $K/\mathbb{Q}$ is not normal: if $\operatorname{Nm}\_{F/\mathbb{Q}}(\delta)=\epsilon^2$ with $\epsilon \in F$, then $(\epsilon/\sqrt{\delta})^2 = \delta'$, implying that all conjugates of $\delta$ are in $K$, contradiction. 
> 


Now, $K'$ contains $\sqrt{\delta \delta'}$, which is a square root of the rational number $a^2-db^2$, so generates a quadratic subextension $F'$, which differs from $F$ precisely because of the claim.


So $\sqrt{-1} \in K'$ is equivalent to $\overline{-1} \in \left\langle \overline{d}, \overline{a^2-db^2} \right\rangle \subset \mathbb{Q}^{\times}/\mathbb{Q}^{\times 2}$.


