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The cyclotomic field $K=\mathbb{Q}(\zeta_{23})$ contains the quadratic field $F=\mathbb{Q}(\sqrt{-23})$, and $F$ has class number $3$ (it is the first quadratic field, when those are ordered by absolu …

The simplest generalisation to abelian surfaces is, I believe, the statement (which is a theorem, not a conjecture) that the Igusa invariants of an abelian surface with CM by $K$ generate an abelian u …

The computation of the Brauer group is only the "first half" of class field theory. The main reason one is interested in the Brauer group is that for a finite Galois extension $L/K$, let's say of loca …

Let $p$ be an odd prime and let $F/\mathbb{Q}$ be a Galois extension with Galois group $D_{2p}$, let $K$ be the intermediate quadratic extension of $\mathbb{Q}$, and $L$ an intermediate degree $p$ ext …

In your particular case, $K^{ab}$ is completely understood, but your field is one of the very few for which such an explicit class field theory is known, so you got lucky. I don't know your backgroun …