Totally imaginary number fields of degree $4$ What are some examples of totally imaginary number fields of degree $4$ that are not a compositum of two imaginary quadratic fields?
How to find such examples (other than by trial and error, considering integer polynomials of degree $4$)?
Is there any useful classification/characterisation of such algebraic number fields? (e.g. in the same way we can say that any totally imaginary number field of degree $2$ is of the form $\mathbb{Q}[\sqrt{d}]$  for some square-free negative integer $d$)?
Thank you.
 A: There are many examples in the number field database https://www.lmfdb.org/NumberField/ . Just type in [0,2] for the signature.
If you want to construct some, take a polynomial of the form
$f(x) = (x^2+10)(x^2+11) + a$ for small values of $a$; its roots will be close to $\sqrt{-10}$ and $\sqrt{-11}$, hence the field generated by a root (if $f$ is irreducible) will be totally complex.
There are classifications of cyclic quartic fields (Kronecker-Weber), and of course you can distinguish the possible Galois groups in the remaining nonabelian cases (dihedral, quaternion, $A_4$, $S_4$ and the Frobenius group), but the Galois group does not fix the form of a generator.
A: Take $K/\mathbb Q$ of degree $4$. If it is Galois, then its Galois group is $\mathbb Z/4\mathbb Z$ or $(\mathbb Z/2 \mathbb Z)^2$. In the first case $K$ is never the compositum of quadratic fields. Example: $\mathbb Q(\exp(2\pi i/5))$. In the second case $K$ is always a compositum of quadratic fields, say $\mathbb Q(\alpha)$ and $\mathbb Q(\beta)$, and if $K$ is imaginary they can be assumed to be imaginary: If one of $\alpha$ or $\beta$ is real, just replace it by $\alpha\beta$.
