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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?

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First, you say that $A_K$ is the minus part of the class group of $K$, which implies that $\mathrm{Gal}(E/E^{+})$ acts on $K$. Yet there is no a priori reason why there should be such an action unless $K = K^{+}.E$ for some cyclic extension $K^{+}/E^{+}$ of degree $p$. (This is the case in your example, but it doesn't seem to be one of your assumptions, so perhaps there are further assumptions you have also omitted, that might be worth checking.) Let's assume the correct assumption is that there exists a cyclic degree $p$ extension $K^{+}/E^{+}$.

The error you point out is indeed an error. To take a very simple example, let $E^{+} = \mathbf{Q}$, let $E$ and $K^{+}$ be quadratic fields, and let $K = E.K^{+}$. (The plus signs are just to correlate with the example above, and are not meant to imply anything about the corresponding fields being real or not.) If $-1$ is a norm from $K$ to $E$, then $N_{E/E^{+}}(-1) = 1$ is also a norm from $K^{+}$ to $E^{+}=\mathbf{Q}$. But there is no reason why the converse should be true. For example, one can take $E$ to be totally real, and $K^{+}$ to be imaginary, so $K/E$ is CM. Then $N_{K/E}(\alpha)$ is always positive. Your example is similar: $N_{E/E^{+}}(\zeta_p) = 1$ if $p$ is odd, and there is no reason why $\zeta_p$ should be a norm in a cyclic extension $K = K^{+}.E$.

I checked: the degree $12$ extension $K/\mathbf{Q}$ had class number $14$, and that the class number of $E$ is $2$. I also checked that $3$ splits in $E/E^{+}$, and that $K/E$ is ramified only at the prime $3$. Thus I agree your counter-example appears to be correct.

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Thanks for your confirmation of your computations in my counterexample and your belief that it meets all the hypotheses I specified. Some notes: (1) stating that $K/E$ is an extension of CM number fields meant that both $K$ and $E$ are CM. Thus, there is a unique totally real subfield of $K$ which we can denote $K^+$. Although I never mentioned $K^+$ in my post, it exists, and the action on $A_K$ is by the Galois group of $K/K^+$. (2) K/E should have degree an odd prime number. In your example, when $K=E$, $[K:E]=1$. If $E$ is totally real and $K$ is totally imaginary, the $[K:E]=2$. – Barry Smith Jan 4 '13 at 0:02
The fact that $K/E$ has odd degree has nothing to do with it; the point is that the "if" statement is false. But thanks for nothing anyway. – user55024 Jan 6 '13 at 0:24
The point is that your purported counterexample did not meet all of the conditions I specified, so is not a counterexample at all. You showed that the "if" statement is false if we weaken the hypotheses and allow $[K:E]$ to be $1$. That does not prove that it is false under the more stringent hypotheses I specified. You did mention the case of interest to me and claim that there is no reason why $\zeta_p$ should be norm, but you didn't explain why. It is just as likely that something special about your example doesn't apply to the case involving $\zeta_p$ as it is that they are analogous. – Barry Smith Jan 8 '13 at 0:20
Confirming that I did find a counterexample is the justification that $\zeta_p$ need not be a norm. So in the end, you answered my second question, and not the first or the third. And once again, I thank you for answering the second question. – Barry Smith Jan 8 '13 at 0:32
up vote 1 down vote accepted

I believe can now answer myself 1), what motivates the statement that "$K/E$ 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 $K^+$"?

If $v$ is a place of $K$, local class field theory gives a reciprocity map from $E_v^{*}$ (abusing notation) to the Galois group of $K_v/E_v$. The kernel consists of the elements that are norms from $K_v^*$. There is a similar map for the localizations of $K^+$ and $E^+$ corresponding to the places below $v$.

The motivation for the claim comes from understanding the relationship between the norm residue map for the localizations of $K^+$ and $E^+$ and the norm residue map for the localizations of $K$ and $E$. A reference is Gras, Class Field Theory, p.79, Corollary 1.5.4. He calls it the "norm lifting theorem".

The problem with the original proof only occurs when the place below $v$ splits in $E/E^+$, since then locally the norm of $\eta$ from $E$ down to $E^+$ is not $1$, it is $\eta$. This is only a problem if the place also ramifies in $K^+/E^+$, since otherwise every unit of $E$, including $\eta$, is a locally a norm down from $K$.

So, the answer to my second question is yes, I did give a true counterexample. A possible answer to the third question is that we require that $E$ not contain a $p$th root of unity. Another answer would be that we require that no place of $E^+$ that splits in $E/E^+$ ramifies in $K^+/E^+$.

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