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Is the following proposition true? If yes, how would you prove this?

Proposition Let $k$ be an algebraic number field. Let $K$ be a finite abelian extension of $k$. Suppose every principal prime ideal of $k$ splits completely in $K$. Let $L$ be a finite extension of $k$. Let $E = KL$. Let $h'$ be the class number of $L$.

Then [$E : L$] | $h'$ and $E/L$ is unramified at every prime ideal of $L$.

Motivation I thought I could use this proposition to prove the following result.

On the class number of a cyclotomic number field of an odd prime order

Effort

Let $\mathcal{I}$ be the group of fractional ideals of $L$. Let $\mathcal{P}$ be the group of principal ideals of $L$. Let $\mathcal{H}$ = {$I \in \mathcal{I}$; $N_{L/k}(I)$ is principal}. Note that $\mathcal{H} \supset \mathcal{P}$. Then use the following two links.

Related questions

On a certain criterion for unramification of an abelian extension of an algebraic number field

Complete splitting of a prime ideal in a certain abelian extension of an algebraic number field

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What is the origin or background of this question? What have you done so far? Is it homework? How far have you got, and where are you stuck? –  Old John Jul 26 '12 at 22:53
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I'm sorry, I down-voted this question as yet-another in a stream of these that strike me as not thought-through much. Only slightly facetiously, I'd wager I could write software to generate a huge number of not-trivial-to-answer questions in alg no th (akin to Gauss' disparagement of Fermat's "Last Theorem"). And, then, as a mathematician, my technical/aesthetic objection is that these questions "have no context", they are just shots-in-the-dark, disconnected. Each reasonable enough in isolation, but... consuming resources? A bit odd, actually. –  paul garrett Jul 26 '12 at 23:52
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@Makato Kato, Yes, indeed, I have noted your many questions of late, and only after these observations do I comment. I am not happy saying negative things, but, if you'll pardon my advice, greater "coherence", or "sense of purpose" would make a better impression on people able to answer your questions. The parallel is not entirely clear, but in English, and several other European languages, there is a fable of "The Boy Who Cried Wolf": it is worth looking up, if one has not ever seen it, and there is a point, that others' interest/care can be worn down... and real issues eventually lost. –  paul garrett Jul 27 '12 at 0:23
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@paulgarett Dear Paul Garett, The motivation for the above question is to find a relation between the class number of a cyclotomic number field and that of its subfield. Please check to see the content of the link in the Motivation section. Regards –  Makoto Kato Jul 27 '12 at 6:11
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@paulgarett Dear Paul Garett, I added Motivation and Effort sections to explain my motivation and how the two links are related to the current question. Regards, –  Makoto Kato Jul 27 '12 at 21:27
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1 Answer

This is a direct application of class field theory. The assumption on $K$ implies that $K$ is contained in the Hilbert class field of $k$. Thus $E$ is contained in the Hilbert class field of $L$, and so $E$ is unramified over $L$, and $[E:L]$ divides the class number of $L$.

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