What is the next important number field? The first few important algebraic number fields I have read about are:


*

*$\mathbb{Q}$ The integers

*$\mathbb{Q}[\sqrt{d}]$ quadratic

*$\mathbb{Q}[e^{\frac{2 i \pi}{p}}]$ cyclotomic


What could be read about next?
 A: Quadratic and cyclotomic fields are important because their structure is simple enough to allow the explicit determintion of some features. In other words, they are a remarkable source of examples.
More intrinsically, cyclotomic fields are important because, by the celebrated Kronecker-Weber theorem, every abelian extension of $\Bbb Q$ is a subextension of a cyclotomic field. An abelian extension is a Galois extension with abelian Galois group.
If we fix a quadratic imaginary field $K$ (i.e. $K={\Bbb Q}(\sqrt{d})$ with $d\in{\Bbb Z}^{<0}$) the theory of complex multiplications tells us where to look for the abelian extensions of $K$. Namely, one considers the complex torus
$$
T=\frac{\Bbb C}{{\Bbb Z}\oplus{\Bbb Z}\tau}
$$
(where $K={\Bbb Q}(\tau)$) which embeds in the projective plane as a non-singular cubic $\cal C$. Then one knows that an abelian extension of $K$ is always a subextension of the field obtained adjoining to $K$ the $x$-coordinate of a point of $\cal C$ image of a point of $T$ of the form $a+b\tau$ with $a$, $b\in{\Bbb Q}$.
This may be a good candidate for the next important (class) of number field(s).
Mind that these are the only established cases where we know explicitly the abelian extensions of a number field.
