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Let me ask a few simple concrete questions (whose answers I’m sure are well known) to motivate my study of class field theory:

What is the maximal abelian unramified extension of $\mathbb{Q}[i]$? (I guess this is called the Hilbert class field).

What is the maximal abelian extension of $\mathbb{Q}[i]$ unramified everywhere except at a prime $p \in \mathbb{Z}$?

Same questions for $\mathbb{Q}[\sqrt{-5}]$ which does not have class number one.

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  • $\begingroup$ $\Bbb Q(i)$ has class number one. $\endgroup$ – Lord Shark the Unknown Oct 12 '18 at 16:25
  • $\begingroup$ Is that three questions now? $\endgroup$ – Lord Shark the Unknown Oct 12 '18 at 16:44
  • $\begingroup$ Ok, I see - subgroups of the class group correspond to abelian extensions so my first question isn’t that interesting. But even in the case of $Q$, the second question has $\mathbb{Q}(\zeta_p)$ as the answer, so if we allow ramification it becomes less trivial. I was choosing imaginary quadratic fields to get complex multiplication (which I don’t yet understand) of elliptic curves answer - there Hilbert class field of $K$ is $K(j(E))$, so what is a concrete example of $K$ in that case? $\endgroup$ – usr0192 Oct 12 '18 at 17:01
  • $\begingroup$ Questions motivate, answers do not. $\endgroup$ – franz lemmermeyer Oct 21 '18 at 10:09
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$\Bbb Q(\sqrt{-5})$ has class number two, so its Hilbert class field is a quadratic extension. That quadratic extension is $\Bbb Q(i,\sqrt5)$.

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