# Examples of unramified abelian extensions of $\mathbb{Q}[i]$

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.

• $\Bbb Q(i)$ has class number one. – Lord Shark the Unknown Oct 12 '18 at 16:25
• Is that three questions now? – Lord Shark the Unknown Oct 12 '18 at 16:44
• 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? – usr0192 Oct 12 '18 at 17:01
• Questions motivate, answers do not. – franz lemmermeyer Oct 21 '18 at 10:09

$$\Bbb Q(\sqrt{-5})$$ has class number two, so its Hilbert class field is a quadratic extension. That quadratic extension is $$\Bbb Q(i,\sqrt5)$$.