# 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

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Marcus's Number Fields has a lot of good exercises on $\mathbb{Q}(\sqrt[3]{\alpha})$, with $\alpha$ a cubefree integer; and $\mathbb{Q}(\sqrt{m},\sqrt{n})$, with $m$ and $n$ distinct squarefree integers different from $1$. – Arturo Magidin Mar 3 '11 at 21:22
What makes a number field important? – Qiaochu Yuan Mar 3 '11 at 22:13
I've always been fond of the exercises in Marcus's books, but then I'm not a professional number theorist, so I could be utterly in the wrong. – Arturo Magidin Mar 3 '11 at 22:25
@Arturo: you're worried that you're wrong to be fond of some piece of mathematics? IMO, that is an utterly groundless worry! – Pete L. Clark Mar 4 '11 at 5:18
@Arturo: well, Marcus's book was one of the first I read, and I am now a number theorist. It's a classical, core-minded text, but there's nothing wrong with that. Anyone interested in number theory will need to read many texts to get a halfway decent picture of the subject... – Pete L. Clark Mar 4 '11 at 7:17

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 $a$, $b\in{\Bbb Q}$.