Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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
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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.