I'm reading a paper that purports to prove the proposition:
Let $K/E$ be a cyclic extension of CM number fields of degree p (an odd prime number). Let $G$ be the Galois group. Let $t$ be the number of primes that split in $E/E^+$ and ramify in $K/E$. Let $A_K$ be the minus part of the $p$-primary class group of $K$, and define $A_E$ similarly. Then
$ \left| A_K^G \right| = \left| A_E \right| \cdot p^t $
The problem is, I seem to have found a counterexample (described at the bottom).
I think I've isolated the issue to the following part of the proof: the authors claim that
$ \left[ \mu_E \colon N_{K/E} K^{*} \cap \mu_E \right] = 1,$
where $\mu_E$ is the group of roots of unity of $E$. They prove this as follows: let $\eta$ be a root of unity in $\mu_E$. The Hasse norm theorem shows one only need show that $\eta$ is locally a norm in $K/E$. They claim that this follows because "$K/E$ is induced from the cyclic extension $K^+/E^+$ by composing with $E$. Thus, locally, $\eta$ is a norm from $K$ if and only if $N_{E/E^+} \eta$ is a norm from $K^+$."
My first question is, what motivates the "if" part of the second sentence?
Now here's what I think is a counterexample: let $E$ be the biquadratic field $\mathbb{Q} (\sqrt{15}, \sqrt{-3})$, and let $K$ be the cubic extension of $E$ generated by a root of the polynomial $x^3-(104958+25725 \sqrt{15}) x + (4821894+1237201 \sqrt{15}).$
The following were determined using PARI/GP: $K/E$ is cyclic. Furthermore, it is ramified only at $p=3$ but $K$ is not the field $E (\zeta_9)$. The prime of $E^+$ dividing $3$ splits in $E/E^+$ and ramifies in $K/E$. Thus, $t=1$. However, $3$ does not divide the class numbers of either $K$ or $E$, so $\left| A_K^G \right| = \left| A_E \right| = 1$. Thus, the proposition appears to be false.
The claim about the group index equalling 1 also seems to be false. PARI/GP indicates that the primitive cube roots of unity in E are not norms of elements in $K^{*}$.
Is this a real counterexample, or am I missing something? If it is, then my third question: is there a simple additional hypothesis under which the Lemma would be true?