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?
The first few important algebraic number fields I have read about are:
What could be read about next?
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.