TITLE: What do we know about the class group of cyclotomic fields over $\mathbb{Q}$?
QUESTION [17 upvotes]: 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:

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?
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!

REPLY [6 votes]: Take the $37^{\text{th}}$ cyclotomic ring of integers, the homomorphism $\sigma:\zeta\to\zeta^2$ with the primitive root $2$ modulo $37$ and the cylotomic integer $g(\zeta)=-\zeta^{13}+\zeta^{25}+\zeta^{27}+\zeta^{30}+\zeta^{32}$. The ideal $\langle g(\zeta)\rangle$ factorizes to $\langle g(\zeta)\rangle=\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle^{2}\cdot\langle149, \sigma^{11}\psi_{149}(\zeta)\rangle$. The second generator $\sigma^{0}\psi_{149}(\zeta)$ of the prime ideal can be taken from the first theorem in chapter $4.12$ of Edwards in Fermat's last theorem. In Dedekind domains it is then easy to prove that the greatest common divisor of the ideals $\langle 149\rangle$ and $\langle\sigma^{0}\psi_{149}(\zeta)\rangle$ is a prime ideal $\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle=\langle 149\rangle+\langle\sigma^{0}\psi_{149}(\zeta)\rangle$ lying over the prime integer $149$. The process of factorizing cyclotomic ideals is described in the chapters $4.11$ff. of Edwards. I outlined this process in my answer to this question. The cyclotomic integer $$h(\zeta)=\zeta-\zeta^{4}-\zeta^{7}-\zeta^{8}-\zeta^{9}-\zeta^{11}-\zeta^{12}-\zeta^{13}-\zeta^{14}-\zeta^{15}+\zeta^{16}-\zeta^{18}-\zeta^{19}+\zeta^{21}-\zeta^{22}-\zeta^{23}-\zeta^{24}-\zeta^{25}-\zeta^{26}-\zeta^{28}-\zeta^{29}-\zeta^{30}-\zeta^{33}+\zeta^{36}$$ factorizes to $\langle h(\zeta)\rangle=\langle149, \sigma^{11}\psi_{149}(\zeta)\rangle\cdot\langle149, \sigma^{29}\psi_{149}(\zeta)\rangle$. The two factorizations give the class equivalence $\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle^{2}\sim\langle149, \sigma^{29}\psi_{149}(\zeta)\rangle$. We square this equivalence and get
$${\left\langle149, \sigma^{0}\psi_{149}(\zeta)\right\rangle^{4}}\sim\left\{\left\langle149, \sigma^{0}\psi_{149}(\zeta)\right\rangle^{2}\right\}^2\sim
{\left\langle149, \sigma^{29}\psi_{149}(\zeta)\right\rangle^2}\sim
{\sigma^{29}\circ\left\langle149, \sigma^{29}\psi_{149}(\zeta)\right\rangle}\sim
{\left\langle149, \sigma^{2\cdot29}\psi_{149}(\zeta)\right\rangle}.$$
Consecutively squaring gives
$$\tag{1}\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle^{2^k}\sim\langle149, \sigma^{29k}\psi_{149}(\zeta).$$
With $\sigma^{36}=\sigma^0\equiv\text{id}$ we get $\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle^{2^{36}}\sim\langle149, \sigma^{0}\psi_{149}(\zeta)$. We take the letter I for the principal class and finally obtain $\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle^{2^{36}-1}\sim I$. Therefore the class order of the prime ideal $\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle$ divides the integer $2^{36}-1$.
Now we take the cyclotomic integer $s(\zeta)=\zeta-\zeta^{4}-\zeta^{17}+\zeta^{25}-\zeta^{27}-\zeta^{34}$. It factorizes to $\langle s(\zeta)\rangle=\langle149, \sigma^{12}\psi_{149}(\zeta)\rangle\cdot\langle149, \sigma^{26}\psi_{149}(\zeta)\rangle\cdot\langle149, \sigma^{27}\psi_{149}(\zeta)\rangle$. With $24\cdot29\equiv12\pmod{36}$ we first get $\sigma^{24\cdot29}=\sigma^{12}$ and equivalence (1) gives the equivalence $\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle^{2^{24}}\sim\langle149, \sigma^{12}\psi_{149}(\zeta)\rangle$. Similarly we get $\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle^{2^{22}}\sim\langle149, \sigma^{26}\psi_{149}(\zeta)\rangle$ and $\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle^{2^{27}}\sim\langle149, \sigma^{27}\psi_{149}(\zeta)\rangle$. This yields
$${\left\langle149, \sigma^{12}\psi_{149}(\zeta)\right\rangle\cdot\left\langle149, \sigma^{26}\psi_{149}(\zeta)\right\rangle\cdot\left\langle149, \sigma^{27}\psi_{149}(\zeta)\right\rangle}\sim
{\left\langle149, \sigma^{0}\psi_{149}(\zeta)\right\rangle^{2^{24}}\cdot\left\langle149, \sigma^{0}\psi_{149}(\zeta)\right\rangle^{2^{22}}\cdot\left\langle149, \sigma^{0}\psi_{149}(\zeta)\right\rangle^{2^{27}}}\sim 
{\left\langle149, \sigma^{0}\psi_{149}(\zeta)\right\rangle^{2^{24}+2^{22}+2^{27}}}\sim I.$$
Hence the class order of the prime ideal $\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle$ divides the integer $2^{24} + 2^{22} + 2^{27}$. But then the class order of the prime ideal $\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle$ divides the greatest common divisor $\gcd (2^{36}-1, 2^{24} + 2^{22} + 2^{27}) = 37$. Kummer's class number formula just gives the prime $37$! We finally get $\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle^{2^{5}}\sim\langle149, \sigma^{0}\psi_{149}(\zeta)\rangle^{32}\sim\langle149, \sigma\psi_{149}(\zeta)\rangle$ with $5\cdot29\equiv1\pmod{36}$. The integer $\xi=2^5=32$ is a primitive root modulo $37$. Therefore there are $36$ prime ideals
$$\tag{2}\sigma^n\left\langle149, \sigma^0\psi_{149}(\zeta)\right\rangle=\left\langle149, \sigma^n\psi_{149}(\zeta)\right\rangle\sim\left\langle149, \sigma^0\psi_{149}(\zeta)\right\rangle^{32^{n}},\quad n=0,\dots,35,$$
and each occupies a different class of the $36$ non-principal classes.
The exponent $f$ of a prime $q$ modulo the prime $p$ is the least positive integer $f$ such that $q^f\equiv1 \pmod{p}$. Kummer already knew that a prime $q$ with exponent $f$ modulo $p$ factorizes to $e=\frac{p-1}{f}$ prime ideals $\langle q, \sigma^k\psi_{q}(\zeta)\rangle$,$0\le k<e$, with $\langle q, \sigma^k\psi_{q}(\zeta)\rangle=\langle q, \sigma^{k+e}\psi_{q}(\zeta)\rangle$ (see chapter $4.9$ of Edwards). Then all prime ideals lying over a prime $q$ with an exponent $f>1$ must be principal because otherwise the homomorphisms $\sigma^0,\sigma^e,\sigma^{2e},\dots,\sigma^{e(f-1)}$ would send the prime ideal
$${\left\langle q, \sigma^{0e}\psi_{q}(\zeta)\right\rangle}=
{\left\langle q, \sigma^{e}\psi_{q}(\zeta)\right\rangle}=
{\left\langle q, \sigma^{2e}\psi_{q}(\zeta)\right\rangle}=\dots=
{\left\langle q, \sigma^{(f-1)e}\psi_{q}(\zeta)\right\rangle}$$
to distinct classes because the primitive root $\xi$=32 has order $36$ modulo $37$. Therefore only prime ideals lying over a prime q with order $f=1$ can be non-principal. If a prime ideal is not principal one conjugate prime ideal $\langle q, \sigma^{k}\psi_{q}(\zeta)\rangle$ must be equivalent to the prime ideal $\langle 149, \sigma^{0}\psi_{149}(\zeta)\rangle$ and the class group is well understood.
Circle divisions and the drawing of regular polygons was very popular in $ 19^{\text{th}} $ century mathematics. Many great names are attached to it. Jacobi (and independently Gauss and Cauchy) found the cyclotomic integer $ \omega_{q,i}(\zeta) $, $ q\equiv1\pmod{p} $, the so-called Jacobi cyclotomic  function, that only factorizes to prime ideals lying over the prime $ q $ and it factorizes either to the prime ideal $ \langle q,\sigma^j\psi_q(\zeta)\rangle $ or the prime ideal $ \langle q,\sigma^{j+\mu}\psi_q(\zeta)\rangle $ with $ \mu=(p-1)/2 $ but it does not factorize to both. Kummer had the idea of choosing exponents $ z_i\in\mathbb{Z} $ such that
\begin{equation*}
\left\lbrace q\cdot\dfrac{\langle q,\sigma^0\psi_q(\zeta)\rangle}{\langle q,\sigma^{\mu}\psi_q(\zeta)\rangle}\right\rbrace^{h_0}=\left\langle\prod_i\omega_{q,i}^{z_i}(\zeta)\right\rangle,\quad h_0\in\mathbb{N}_0.
\end{equation*}
If the prime ideal $ \langle q,\sigma^0\psi_q(\zeta)\rangle $ is principal, there exists a cyclotomic integer $ g(\zeta) $ such that
\begin{equation*}
\left\langle g(\zeta)\right\rangle=q\cdot\dfrac{\langle q,\sigma^0\psi_q(\zeta)\rangle}{\langle q,\sigma^{\mu}\psi_q(\zeta)\rangle}
\end{equation*}
with $ \langle q,\sigma^{\mu}\psi_q(\zeta)\rangle\mid \langle q\rangle $ and there exists a unit $ \varepsilon $ such that
\begin{equation*}
\varepsilon\cdot g^{h_0}(\zeta)=\prod_i\omega_{q,i}^{z_i}(\zeta).
\end{equation*}
Kummer could compute this unit to $ \varepsilon=(-\zeta)^k $ with some integer $ k $. We apply another method of Kummer, that he used in order to prove that the class number of the  $37^{\text{th}} $ real cyclotomic ring of integers is not divisible by the prime $ p=37 $, and take the power of $ n $ such that $ (-\zeta)^{kn}=1 $ and $ h_0n=r-1 $ hold for some prime $ r\equiv1\pmod{p} $. By group theory we have
\begin{equation*}
\left\lbrace (-\zeta)^k\cdot g^{h_0}(\zeta)\right\rbrace^n=g^{h_0n}(\zeta)=g^{r-1}(\zeta)\equiv1\pmod{\langle r,\sigma^j\psi_r(\zeta)\rangle}
\end{equation*}
for any conjugate prime ideal $ \langle r,\sigma^j\psi_r(\zeta)\rangle $ in the multiplicative group of the quotient field $ \mathcal{O}_{\mathbb{Q}[\zeta]}/\langle r,\sigma^j\psi_r(\zeta)\rangle $ with order $ r-1 $. Now, if
\begin{equation*}
\left\lbrace\prod_i\omega_{q,i}^{z_i}(\zeta)\right\rbrace^n\not\equiv1\pmod{\langle r,\sigma^j\psi_r(\zeta)\rangle}
\end{equation*}
the prime ideal $ \langle q,\sigma^j\psi_q(\zeta)\rangle $ cannot be principal. Incongruence is given for $ h_0=37 $, $ q=149 $, $ r=5477=4\cdot37^2+1 $ and every conjugate prime ideal $ \langle r,\sigma^j\psi_r(\zeta)\rangle $.
A detailed approach to these mysterious determinants $ h_0 $, as Kummer and Gauss put it, can be taken from section $ 6.10 $, here. Chapter $ 21 $ contains the class groups of all $ p^{\text{th}} $ cyclotomic rings of integers for odd primes $ p\leq71 $.