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As KConrad says, the answer to the first question is "not yet known". For Iwasawa theory, you could consider the extension $L_\infty=\mathbb{Q}(\sqrt[p^\infty]{2},\zeta_{p^\infty})$ which is Galois ov …

As Franz Lemmermeyer suggested, one should consider the so-called Ambigous Class Number Formula: you find it, for instance, in Gras' book "Class Field Theory", II.6.2.3. It says that if $L/K$ is a fi …

I have been trying to prove the result for a couple of days, without success, so I post what I got in the meanwhile. Let me suppose throughout that $\operatorname{Gal}(E/K)\cong(\mathbb{Z}/p)^2$ (the …

In German Strahl means ray but is mathematically used to mean a half-line, infinite in only one direction. I guess that the use of Strahlklassengruppe refers to the fact that the ray class field modul …

As Keith Conrad observed in his comment, what you are asking about is Washington's theorem which is either in his book, referenced as Theorem 16.12, or in his original paper in Invent. Math. He proved …

It is not very clear to me what you mean by "intersection of Hilbert Class Fields [...] is discussed". The theory of Complex Multiplication (see, for instance, Serre's short note in Cassels and Frohli …

I normally don't like to cite my own work on MO, but this time the preprint arXiv:1803.04064 was written, together with L. Caputo, having the OP's question in mind; and so, first of all, let me thank …

I think that the point lies in the difference between a primitive and imprimitive $L$-function. Before entering the details, let me observe that Washington's definition of $p$-adic $L$-functions (as t …

The answer to both my question is that "adding conductors does not change anything". Olivier has already discussed this for the Principal Ideal Theorem, and for Hilbert 94 this is proven by Suzuki in …

First of all, as they observe, the assumption that $H(K)/\mathbb{Q}$ is abelian ensures that $H(K)\subseteq\mathbb{Q}(\zeta_{f(K)})$ and not only $K\subseteq\mathbb{Q}(\zeta_{f(K)})$. Also, they work …

Well, actually the kernel of $\theta$ is perfectly explicit and it is the connected component of the identity in $C_K$: see, for instance, Artin-Tate Class Field Theory, Chapter IX, §1. Theorem 3 ibid …

Let $L/K$ be a finite, cyclic extension of number fields, say with $\mathrm{Gal}(L/K)=G$. In my context $G$ is actually of order $p$, an odd prime number, but let me state my question for every cyclic …

Given your assumption that $E/K$ has CM by $\mathcal{O}_K$, it follows that $K$ has class numer $1$. In particular, there are isomorphisms $\mathrm{Gal}(E[\mathfrak{a}])/K\cong\mathrm{Gal}(K(\mathfrak …

I don't think we know how to prove this directly. Indeed, recent works by Wake and Wake–Erickson show that this cyclicity is equivalent to a conjectured improvement of Mazur–Wiles' result to the effec …

Have you had a look at Schoof's paper The structure of the minus class group of abelian number fields, Séminaire de Théorie des Nombres de Paris, 1988--89? Schoof's idea is to conjecture that the Fitt …