Motivated by this question, I am curious how one can characterize primes that splits completely in the Hilbert class field of $\mathbb{Q}(\zeta_q)$, where $q$ is a prime. Then I realize how much I don't know about this extension over $\mathbb{Q}$! I know this is too broad, but I'll be happy if someone can briefly write down what we can say about this extension in general. At the very least
Is the extension $H/\mathbb{Q}$ abelian, if $H$ is the Hilbert class field of $\mathbb{Q}(\zeta_q)$, where $q$ is a prime? I guess it's generally not. If so, when is it abelian?
and
What is the degree of $H/\mathbb{Q}$? What do we know about the ideal class groups of $\mathbb{Q}(\zeta_q)$?
Actually, I am not even sure what the right question to ask here is. A quick google search yields some relevant things, such as this MO post and Franz Lemmermeyer's papers.. They seem to say that:
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We can't pin down the Hilbert class field of cyclotomic field unless it's imaginary quadratic or $\mathbb{Q}$ itself. But can we say anything about whether it's abelian, say?
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There is a connection to the main theorem of Iwasawa theory. I know that this is too demanding, but if someone's willing to sketch a quick introduction to Iwasawa theory and how it connects to this problem here, I would be super grateful.
Thanks!
Best Answer
The extension $H/\mathbf Q$ is never abelian, unless $H=\mathbf Q(\zeta_q)$ (which happens for just a few values of $q$, since the class number of $\mathbf Q(\zeta_n)$ grows very fast). Indeed, according to Kronecker-Weber, every abelian extension of $\mathbf Q$ is contained in a cyclotomic extension. If $H/\mathbf Q$ were abelian, we would have $\mathbf Q(\zeta_q) \subseteq H \subseteq \mathbf Q(\zeta_n)$ for some $n$; then, examining the ramification, we see that $n=q$ and so $H=\mathbf Q(\zeta_q)$.
The degree of $H/\mathbf Q$ is equal to $q-1$ times the class number $h_q$.
In general, the primes which split completely in the Hilbert class field are those which are principal in the base field.
The class group of $\mathbf Q(\zeta_n)$ is a very complicated thing. Whole books are devoted to it. It is a fascinating topic with deep connections to the theory of the Riemann zeta function.
I recommend Washington's book Introduction to Cyclotomic Fields. If you are interested specifically in Iwasawa theory, I would recommend the book Cyclotomic Fields and Zeta Values by Coates and Sujatha, which is a great introduction to this truly unbelievable theory.