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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?

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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
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What makes a number field important? –  Qiaochu Yuan Mar 3 '11 at 22:13
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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
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@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
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@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
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up vote 5 down vote accepted

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.

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