IDEAL CLASS GROUPS OF CYCLOTOMIC NUMBER FIELDS I

Franz Lemmermeyer

Institut für Mathematik Universität Heidelberg

July 1994

Abstract. Following Hasse’s example, various authors have been deriving divisibility properties of minus class numbers of cyclotomic fields by carefully examining the analytic class number formula. In this paper we will show how to generalize these results to CM-fields by using class field theory. Although we will only need some special cases, we have also decided to include a few results on Hasse’s unit index for CM-fields as well, because it seems that our proofs are more direct than those in Hasse’s book [H].

1. Notation

Let K ⊂ L be number fields; we will use the following notation:

𝔒_{K} is the ring of integers of K;

E_{K} is its group of units; W_{K} is the group of roots of unity contained in K; w_{K} is the order of w_{K};

Cl(K) is the ideal class group of K;

K^{1} denotes the Hilbert class field of K, that is the maximal abelian extension

of K which is unramified at all places; j_{K→L} denotes the transfer of ideal classes for number fields K ⊂ L, i.e. the

homomorphism Cl(K) → Cl(L) induced by mapping an ideal 𝔞 to 𝔞𝔒_{L}; κ_{L/K} denotes the capitulation kernel ker j_{K→L}; Now let K be a CM-field, i.e. a totally complex quadratic extension of a totally real number field; the following definitions are standard:

σ is complex conjugation; K^{+} denotes the maximal real subfield of K; this is the subfield fixed by σ.

Cl^{−}(K) is the kernel of the map N_{K/K^{+}} : Cl(K) → Cl(K^{+}) and is called the minus

class group; h^{−}(K) is the order of Cl^{−}(K), the minus class number; Q(K) = (E_{K} : W_{K}E_{K^{+}}) ∈ {1, 2} is Hasse’s unit index.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary 11 R 18; Secondary 11 R 29.

Key words and phrases. Ideal class group, cyclotomic fields, unit index, class number formula.

We will need a well known result from class field theory. Assume that K ⊂ L are CM-fields; then ker(N_{L/K} : Cl(L) → Cl(K)) has order (L ∩ K^{1} : K). Since K/K^{+} is ramified at the infinite places, the norm N_{K/K^{+}} : Cl(K) → Cl(K^{+}) is onto.

2. Hasse’s unit index

Hasse’s book [H] contains numerous theorems (Sätze 14 – 29) concerning the unit index Q(L) = (E_{L} : W_{L}E_{K}), where K = L^{+} is the maximal real subfield of a cyclotomic number field L. Hasse considered only abelian number fields L/ℚ, hence he was able to describe these fields in terms of their character groups X(L); as we are interested in results on general CM-fields, we have to proceed in a different manner. But first we will collect some of the most elementary properties of Q(L) (see also [H] and [W]) in

Proposition 1. Let K ⊂ L be CM-fields; then a) Q(L) = (E_{L} : W_{L}E_{K}) = (E_{L}^{σ−1} : W_{L}^{2}) = (E_{L}^{σ+1} : E_{K}^{2});

b) Q(L) ∈ {1, 2}, and if Q(L) = 2 then κ_{L/L^{+}} = 1;

c) (Hasse’s Satz 29) Q(K)|Q(L) · (W_{L} : W_{K});

d) (compare Hasse’s Satz 26) Suppose that N_{L/K} : W_{L}/W_{L}^{2} → W_{K}/W_{K}^{2} is onto (for example, assume that w_{L} ≡ 2 mod 4 and that (L : K) is odd). Then Q(L) = 2 implies Q(K) = 2.

e) If L = ℚ(ζ_{m}), where m ≢ 2 mod 4 is composite, then Q(L) = 2;

f) Let K_{1} ⊆ ℚ(ζ_{m}) and K_{2} ⊆ ℚ(ζ_{n}) be abelian CM-fields, where m = p^{μ} and n = q^{ν} are prime powers such that p ≠ q, and let K = K_{1}K_{2}; then Q(K) = 2.

The proofs are straight forward:

a) σ − 1 induces an isomorphism E_{L}/W_{L}E_{K} ⟶ E_{L}^{σ−1}/W_{L}^{2}, hence we have (E_{L} : W_{L}E_{K}) = (E_{L}^{σ−1} : W_{L}^{2}). The other claim is proved similarly.

b) Since W_{L}/W_{L}^{2} is cyclic of order 2, the first claim follows immediately from a). Now let 𝔞 be an ideal in 𝔒_{K} such that 𝔞𝔒_{L} = α𝔒_{L}. Then α^{σ−1} = ζ for some root of unity ζ ∈ L, and Q(L) = 2 shows that ζ = ε^{σ−1} for some ε ∈ E_{L}. Now αε^{−1} generates 𝔞 and is fixed by σ, hence lies in K. This shows that 𝔞 is principal in K, i.e. that κ_{L/L^{+}} = 1.

c) First note that (W_{L} : W_{K}) = (W_{L}^{2} : W_{K}^{2}); then

Q(L) · (W_{L} : W_{K}) = (E_{L}^{σ−1} : W_{L}^{2})(W_{L}^{2} : W_{K}^{2})
= (E_{L}^{σ−1} : E_{K}^{σ−1})(E_{K}^{σ−1} : W_{K}^{2})
= (E_{L}^{σ−1} : E_{K}^{σ−1}) · Q(K)

proves the claim.

d) Since Q(L) = 2, there is a unit ε ∈ E_{L} such that ε^{σ−1} = ζ generates W_{L}/W_{L}^{2}. Taking the norm to K shows that (N_{L/K}ε)^{σ−1} = N_{L/K}(ζ) generates W_{K}/W_{K}^{2}, i.e. we have Q(K) = 2.

e) In this case, 1 − ζ_{m} is a unit, and we find (1 − ζ_{m})^{1−σ} = −ζ_{m}. Since −ζ_{m} ∈ W_{L} \ W_{L}^{2}, we must have Q(L) = 2; f) Assume that m and n are odd. A subfield K ⊆ L = ℚ(ζ_{m}), where m = p^{μ}

is an odd prime power, is CM if and only if it contains the maximal 2extension contained in L, i.e. if and only (L : K) is odd. Since L = ℚ(ζ_{mn}) has unit index Q(L) = 2, the assertion follows from d) and e).

Now assume that p = 2. If w_{L} ≡ 2 mod 4, we can employ d) to prove the claim. We may therefore assume that \sqrt{−1} ∈ L. In this case, we must have K_{1} = ℚ(ζ_{m}) for m = 2^{α} and some α ≥ 2 (complex subfields of the field of 2^{μ}th roots of unity containing \sqrt{−1} necessarily have this form). Now n is odd and K_{2} ⊆ ℚ(ζ_{n}) is complex, hence (ℚ(ζ_{n}) : K_{2}) is odd. The proof of d) shows that it suffices to show that K_{1}(ζ_{n}) has unit index 2, and this is just the assertion of e). □

Now let L be a CM-field with maximal real subfield K; we will call L/K essentially ramified if L = K(\sqrt{α}) and there is a prime ideal 𝔭 in 𝔒_{K} such that 𝔭 ∥α (this is to say that 𝔭|α𝔒_{K}, but 𝔭^{2} ∤ α𝔒_{K}); it is easily seen that this does not depend on which α we choose. We leave it as an exercise to the reader to verify that our definition of essential ramification coincides with Hasse’s [H, Sect. 22]; the key observation is the ideal equation (4α) = 𝔞^{2}𝔡, where 𝔡 = diff (K(\sqrt{α})/K).

We will also need certain totally real elements of norm 2 in the field of 2^{m}th roots of unity: to this end we define

π_{2} = 2 = 2 + ζ_{4} + ζ_{4}^{−1},
π_{3} = 2 + \sqrt{2} = 2 + ζ_{8} + ζ_{8}^{−1},
π_{4} = 2 + \sqrt{2 + \sqrt{2}} = 2 + ζ_{16} + ζ_{16}^{−1},
…,
π_{n} = 2 + \sqrt{π_{n−1}} = 2 + ζ_{2^{n}} + ζ_{2^{n}}^{−1}.

Let m ≥ 2, L = ℚ(ζ_{2^{m+1}}) and K = ℚ(π_{m}); then L/K is an extension of type (2, 2) with subfields K_{1} = ℚ(ζ_{2^{m}})/K, K_{2} = ℚ(\sqrt{π_{m}}) and K_{3} = ℚ(\sqrt{−π_{m}}). Moreover, K_{2}/K and K_{3}/K are essentially ramified, whereas K_{1}/K is not.

Theorem 1. Let L be a CM-field with maximal real subfield K; we will distinguish several cases:

1. w_{L} ≡ 2 mod 4 1. If L/K is essentially ramified, then Q(L) = 1, and κ_{L/K} = 1.

2. L/K is not essentially ramified. Then L = K(\sqrt{α}) for some α ∈ 𝔒_{K} such that α𝔒_{K} = 𝔞^{2}, where 𝔞 is an integral ideal in 𝔒_{K}. Now

a) Q(L) = 2, if 𝔞 is principal, and

b) Q(L) = 1 and κ_{L/K} = ⟨[𝔞]⟩, if 𝔞 is not principal.

2. w_{L} ≡ 2^{m} mod 2^{m+1}, where m ≥ 2. Then L/K is not essentially ramified,

and 1. if π_{m}𝔒_{K} is not an ideal square, then Q(L) = 1 and κ_{L/K} = 1;

2. if π_{m}𝔒_{K} = 𝔟^{2} for some integral ideal 𝔟, then a) Q(L) = 2, if 𝔟 is principal, and

b) Q(L) = 1, κ_{L/K} = ⟨[𝔟]⟩, if 𝔟 is not principal.

For the proof of Theorem 1 we will need the following Lemma 1. Let L = K(\sqrt{π}), and let σ denote the non-trivial automorphism of L/K. Moreover, let 𝔟 be an ideal in 𝔒_{K} such that 𝔟𝔒_{L} = (β) and β^{σ−1} = −1 for some β ∈ L. Then π𝔒_{K} is an ideal square in 𝔒_{K}.

If, on the other hand, β^{σ−1} = ζ, where ζ is a primitive 2^{m}th root of unity, then π_{m}𝔒_{K} is an ideal square in 𝔒_{K}.

Proof. We have (β\sqrt{π})^{σ−1} = 1, hence β\sqrt{π} ∈ K. Therefore 𝔟 and 𝔠 = (β\sqrt{π}) are ideals in 𝔒_{K}, and (𝔠𝔟^{−1})^{2} = π𝔒_{K} proves our claim.

Now assume that β^{σ−1} = ζ; then σ fixes (1 − ζ)β^{−1}, hence ((1 − ζ)β) and 𝔠 = (1−ζ) = 𝔠 are ideals in 𝔒_{K}, and 𝔠^{2} = N_{L/K}(1−ζ) = (2+ζ +ζ^{−1})𝔒_{K} is indeed an ideal square in 𝔒_{K} as claimed. □

Proof of Theorem 1. There are the following cases to consider:

1. w_{L} ≡ 2 mod 4

1. L/K is essentially ramified.

Assume we had Q(L) = 2; then E_{L}^{σ−1} = W_{L}, hence there is a unit ε ∈ E_{L} such that ε^{σ−1} = −1. Write L = K(\sqrt{π}), and apply Lemma 1 to 𝔟 = (1), β = ε: this will yield the contradiction that L/K is not essentially ramified.

2. L/K is not essentially ramified.

Then L = K(\sqrt{α}) for some α ∈ 𝔒_{K} such that α𝔒_{K} = 𝔞^{2}, where 𝔞 is an integral ideal in 𝔒_{K}.

a) If 𝔞 is principal, say 𝔞 = β𝔒_{K}, then there is a unit ε ∈ E_{K} such that α = β^{2}ε, and we see that L = K(\sqrt{ε}). Now \sqrt{ε}^{σ−1} = −1 is no square since w_{L} ≡ 2 mod 4, and Prop. 1.a) gives Q(L) = 2.

b) If 𝔞 is not principal, then the ideal class [𝔞] capitulates in L/K because 𝔞𝔒_{L} = \sqrt{α}𝔒_{L}. Prop. 1.b) shows that Q(L) = 1.

2. w_{L} ≡ 2^{m} mod 2^{m+1} for some m ≥ 2

1. If we have Q(L) = 2 or κ_{L/K} ≠ 1, then Lemma 1 says that π_{m}𝔒_{K} = 𝔟^{2} is an ideal square in 𝔒_{K} in contradiction to our assumption.

2. Suppose therefore that π_{m} = 𝔟^{2} is an ideal square in 𝔒_{K}. If 𝔟 is not principal, then 𝔟𝔒_{L} = (1−ζ) shows that κ_{L/K} = ⟨[𝔟]⟩, and Prop. 1.b) gives Q(L) = 1. If, on the other hand, 𝔟 = β𝔒_{K}, then ηβ^{2} = π_{m} for some unit η ∈ E_{K}. If η were a square in 𝔒_{K}, then π_{m} would also be a square, and L = K(\sqrt{−1} ) would contain the 2^{m+1}th roots of unity. Now ηβ^{2} = π_{m} = ζ^{−1}(1 + ζ)^{2}, hence ηζ is a square in L, and we have Q(L) = 2 as claimed. □

Remark. For L/ℚ abelian, Theorem 1 is equivalent to Hasse’s Satz 22; we will again only sketch the proof: suppose that w_{L} ≡ 2^{m} mod 2^{m+1} for some m ≥ 2, and define L′ = L(ζ_{2^{m+1}}), K′ = L′ ∩ ℝ. Then K′/K is essentially ramified if and only if π_{m−2} is not an ideal square in 𝔒_{K} (because K′ = K(π_{m+1}) = K(\sqrt{π_{m}})). The asserted equivalence should now be clear.

Examples.

1. Complex subfields L of ℚ(ζ_{p^{m}}), where p is prime, have unit index Q(L) = 1 (Hasse’s Satz 23) and κ_{L/L^{+}} = 1: since p ramifies completely in ℚ(ζ_{p^{m}})/ℚ, L/L^{+} is essentially ramified if p ≠ 2, and the claim follows from Theorem 1.

If p = 2 and L/L^{+} is not essentially ramified, then we must have L = ℚ(ζ_{2^{μ}}) for some μ ∈ ℕ, and we find Q(L) = 1 by Theorem 1.2.1.

2. If K is a CM-field, which is essentially ramified at a prime ideal 𝔭 above p ∈ ℕ, and if F is a totally real field such that p ∤ disc F, then Q(L) = 1 and κ_{L/L^{+}} = 1 for L = KF: this is again due to the fact that either L/L^{+} is essentially ramified at the prime ideals above 𝔭, or p = 2 and K = K^{+}(\sqrt{−1} ). In the first case, we have Q(L) = 1 by Theorem 1.1.1, and in the second case by Theorem 1.2.1.

3. (compare Prop. 1.f)) Let K ⊆ ℚ(ζ_{m}) be an abelian CM-field, suppose that F ⊆ ℚ(ζ_{n}) ∩ ℝ is totally real (here m = p^{μ} and n = q^{ν} are prime powers such that p ≠ q), and let L = KF; then Q(L) = 2. This is in fact a special case of example 2.

4. Cyclic extensions L/ℚ have unit index Q(L) = 1 (Hasse’s Satz 24): Let F be the maximal subfield of L such that (F : ℚ) is odd. Then F is totally real, and 2 ∤ disc F (this follows from the theorem of Kronecker and Weber). Similarly, let K be the maximal subfield of L such that (K : ℚ) is a 2power: then K is a CM-field, and L = F K. If K/K^{+} is essentially ramified at a prime ideal 𝔭 above an odd prime p, then so is L/L^{+}, because L/ℚ is abelian, and all prime ideals in F have odd ramification index. Hence the claim follows by example 2. above.

If, however, K/K^{+} is not essentially ramified at a prime ideal 𝔭 above an odd prime p, then disc K is a 2-power (recall that K/ℚ is cyclic of 2-power degree). Applying the theorem of Kronecker and Weber, we find that K ⊆ ℚ(ζ), where ζ is some primitive 2^{m}th root of unity. If K/K^{+} is essentially ramified at a prime ideal above 2, then so is L/L^{+}, and Theorem 1 gives us Q(L) = 1. If K/K^{+} is not essentially ramified at a prime ideal above 2, then we must have K = ℚ(ζ), where ζ is a primitive 2^{m}th root of unity; but now π_{m−2}𝔒_{L^{+}} is not the square of an integral ideal, and we have Q(L) = 1 by Theorem 1.

3. Masley’s theorem h_{m}^{−}|h_{mn}^{−}

Now we can prove a theorem which will contain Masley’s result h^{−}(K) | h^{−}(L) for cyclotomic fields K = ℚ(ζ_{m}) and L = ℚ(ζ_{mn}) as a special case:

Theorem 2. Let K ⊂ L be CM-fields; then

h^{−}(K) | h^{−}(L) · |κ_{L/L^{+}} | · \frac{(L ∩ K^{1} : K)}{(L^{+} ∩ (K^{+})^{1} : K^{+})},

and the last quotient is a power of 2.

Proof. Let ν_{K} and ν_{L} denote the norms N_{K/K^{+}} and N_{L/L^{+}}, respectively; then the following diagram is exact and commutative:

1 ⟶ Cl^{−}(L) ⟶ Cl(L) ⟶^{ν_{L}} Cl(L^{+}) ⟶ 1
↓_{N^{−}}  ↓^{N}  ↓_{N^{+}}
1 ⟶ Cl^{−}(K) ⟶ Cl(K) ⟶^{ν_{K}} Cl(K^{+}) ⟶ 1

The snake lemma gives us an exact sequence

1 → ker N^{−} → ker N → ker N^{+} → cok N^{−} → cok N → cok N^{+} → 1.

Let h(L/K) denote the order of ker N, and let h^{−}(L/K) and h(L^{+}/K^{+}) be defined accordingly. We have seen above that class field theory tells us

| cok N | = (L ∩ K^{1} : K), | cok N^{+} | = (L^{+} ∩ (K^{+})^{1} : K^{+}).

The alternating product of the orders of the groups in exact sequences equals 1, so the above sequence implies

h^{−}(L/K) · h(L^{+}/K^{+}) · |cok N| = h(L/K) · |cok N^{−}| · |cok N^{+}|.

The exact sequence

1 → ker N^{−} → Cl^{−}(L) → Cl^{−}(K) → cok N^{−} → 1

gives us

h^{−}(L/K) · h^{−}(K) = h^{−}(L) · |cok N^{−}|.

Collecting everything we find that

h^{−}(K) · \frac{h(L/K)}{h(L^{+}/K^{+})} · \frac{(L^{+} ∩ (K^{+})^{1} : K^{+})}{(L ∩ K^{1} : K)} = h^{−}(L).

Now the claimed divisibility property follows if we can prove that h(L^{+}/K^{+}) divides h(L/K) · | κ_{L/L^{+}} |. But this is easy: exactly h(L^{+}/K^{+})/| κ_{L/L^{+}} | ideal classes of ker N^{+} ⊂ Cl(L^{+}) survive the transfer to Cl(L), and if the norm of L^{+}/K^{+} kills an ideal class c ∈ Cl(L^{+}), the same thing happens to the transferred class c^{j} when the norm of L/K is applied. We remark in passing that | κ_{L/L^{+}} | ≤ 2 (see Hasse [H, Satz 18]).

It remains to show that (L ∩ K^{1} : K)/(L^{+} ∩ (K^{+})^{1} : K^{+}) is a power of 2. Using induction on (L : K), we see that it suffices to prove: if L/K is an unramified abelian extension of CM-fields of odd prime degree (L : K) = q, then so is L^{+}/K^{+}. Suppose otherwise; then there exists a finite prime 𝔭 which ramifies, and since L^{+}/K^{+} is cyclic, 𝔭 has ramification index q. Now L/K^{+} is cyclic of order 2q, hence K must be the inertia field of 𝔭, contradicting the assumption that L/K be unramified. We conclude that L^{+}/K^{+} is also unramified, and so odd factors of (L ∩ K^{1} : K) cancel against the corresponding factors of (L^{+} ∩ (K^{+})^{1} : K^{+}). □

Corollary. If K = ℚ(ζ_{m}) and L = ℚ(ζ_{mn}) for some m, n ∈ ℕ, then h^{−}(K) | h^{−}(L).

Proof. We have shown in Sect. 2 that j_{K^{+}→K} and j_{L^{+}→L} are injective in this case. Moreover, L/K does not contain a nontrivial subfield of K^{1} (note that p is completely ramified in L/K if n = p, and use induction). □

This Corollary is due to Masley [M]; the special case m = p^{a}, n = p can already be found in [We]. Examples of CM-fields L/K such that h^{−}(K) ∤ h^{−}(L) have been given by Hasse [H]; here are some more:

Examples.

1. Let d_{1} ∈ {−4, −8, −q (q ≡ 3 mod 4)} be a prime discriminant, and suppose that d_{2} > 0 is the discriminant of a real quadratic number field. Then h^{−}(K) ∤ h^{−}(L) for K = ℚ(\sqrt{d_{1}d_{2}} ), L = ℚ(\sqrt{d_{1}}, \sqrt{d_{2}} ). In this example, (L ∩ K^{1} : K) = 2 · (L^{+} ∩ (K^{+})^{1} : K^{+}) and κ_{L/L^{+}} = 1.

2. Let d_{1} = −4, d_{2} = 8m for some odd m ∈ ℕ, and suppose that 2 = (2, \sqrt{2m}) is not principal in 𝔒_{k}, where k = ℚ(\sqrt{2m}). Then h^{−}(K) ∤ h^{−}(L) for K = ℚ(\sqrt{−2m}), L = ℚ(\sqrt{−1}, \sqrt{2m}). Here (L ∩ K^{1} : K) = (L^{+} ∩ (K^{+})^{1} : K^{+}), but κ_{L/L^{+}} = ⟨[2]⟩, since 2𝔒_{L} = (1 + i).

Other examples can be found by replacing d_{1} in Example 2 by d_{1} = −8 or d_{2} by d_{2} = 4m, m ∈ ℕ odd. The proof that in fact h^{−}(K) ∤ h^{−}(L) for these fields uses Theorem 1, as well as Prop. 2 and 3 below.

4. Metsänkylä’s factorization

An extension L/K is called a V_{4}-extension of CM-fields, if 1. L/K is normal and Gal(L/K) ≃ V_{4} = (2, 2);

2. Exactly two of the three quadratic subfields are CM-fields; call them K_{1}

and K_{2}, respectively. This implies, in particular, that K is totally real, and that L is a CM-field with maximal real subfield L^{+} = K_{3}. We will write Q_{1} = Q(K_{1}), W_{1} = W_{K_{1}}, etc.

Louboutin [Lou, Prop. 13] has given an analytic proof of the following class number formula for V_{4}-extension of CM-fields, which contains Lemma 8 of Ferrero [F] as a special case:

Proposition 2. Let L/K be a V_{4}-extension of CM-fields; then

h^{−}(L) = \frac{Q(L)}{Q_{1}Q_{2}} \frac{w_{L}}{w_{1}w_{2}} h^{−}(K_{1})h^{−}(K_{2}).

Proof. Kuroda’s class number formula (for an algebraic proof see [L]) yields

h(L) = 2^{d−κ−2−υ}q(L)h(K_{1})h(K_{2})h(L^{+})/h(K)^{2},  (1)

where d = (K : ℚ) is the number of infinite primes of K which ramify in L/K; κ = d − 1 is the ℤ-rank of the unit group of K; υ = 1 if and only if all three quadratic subfields of L/K can be written as K(\sqrt{ε})

for units ε ∈ E_{K}, and υ = 0 otherwise; q(L) = (E_{L} : E_{1}E_{2}E_{3}) is the unit index for extensions of type (2, 2); here E_{j} is the

unit group of K_{j} (similarly, let W_{j} denote the group of roots unity in L_{j}). Now we need to find a relation between the unit indices involved; we assert

Proposition 3. If L/K is a V_{4}-extension of CM-fields, then

\frac{Q(L)}{Q_{1}Q_{2}} \frac{w_{L}}{w_{1}w_{2}} = 2^{−1−υ}q(L).

Proof of Prop. 3. We start with the observation

Q(L) = (E_{L} : W_{L}E_{3}) = (E_{L} : E_{1}E_{2}E_{3})\frac{(E_{1}E_{2}E_{3} : W_{1}W_{2}E_{3})}{(W_{L}E_{3} : W_{1}W_{2}E_{3})}.

In [L] we have defined groups E_{j}^{∗} = {ε ∈ E_{j} : N_{j}ε is a square in E_{K}}, where N_{j} denotes the norm of K_{j}/K; we also have shown that

(E_{1}E_{2}E_{3} : E_{1}^{∗}E_{2}^{∗}E_{3}^{∗}) = 2^{−υ} ∏(E_{j} : E_{j}^{∗})

and E_{j}/E_{j}^{∗} ≃ E_{K}/N_{j}E_{j}. Now Prop. 1.a) gives (E_{K} : N_{j}E_{j}) = Q_{j} for j = 1, 2, and we claim

1. (W_{L}E_{3} : W_{1}W_{2}E_{3}) = (W_{L} : W_{1}W_{2}) = 2 · \frac{w_{L}}{w_{1}w_{2}} ;

2. E_{1}^{∗}E_{2}^{∗}E_{3}^{∗} = W_{1}W_{2}E_{3}^{∗};
3. (W_{1}W_{2}E_{3} : W_{1}W_{2}E_{3}^{∗}) = (E_{3} : E_{3}^{∗}).

This will give us

Q(L) = 2^{−1−υ}q(L)Q_{1}Q_{2} \frac{w_{1}w_{2}}{w_{L}};

inserting this into equation (1) and recalling the definition of the minus class number yields Louboutin’s formula.

We still have to prove the three claims above:

1. W_{L}E_{3}/W_{1}W_{2}E_{3} ≃ W_{L}/(W_{L} ∩ W_{1}W_{2}E_{3}) ≃ W_{L}/W_{1}W_{2}, and the claim fol-

lows from W_{1} ∩ W_{2} = {−1, +1};

2. We only need to show that E_{1}^{∗}E_{2}^{∗}E_{3}^{∗} ⊂ W_{1}W_{2}E_{3}^{∗}; but Prop. 1.a) shows that

ε ∈ E_{1}^{∗} ⟺ ε^{σ+1} ∈ E_{K}^{2} ⟺ ε ∈ W_{1}E_{K}, and this implies the claim;

3. W_{1}W_{2}E_{3}/W_{1}W_{2}E_{3}^{∗} ≃ E_{3}/E_{3} ∩ W_{1}W_{2}E_{3}^{∗} ≃ E_{3}/E_{3}^{∗}. □

Combining the result of Section 3 with Prop. 2, we get the following

Theorem 3. Let L_{1} and L_{2} be CM-fields, and let L = L_{1}L_{2} and K = L_{1}^{+}L_{2}^{+}; then L/K is a V_{4}−extension of CM-fields with subfields K_{1} = L_{1}L_{2}^{+}, K_{2} = L_{1}^{+}L_{2}, K_{3} = L^{+}, and

h^{−}(L) = \frac{Q(L)}{Q_{1}Q_{2}} \frac{w_{L}}{w_{1}w_{2}} h^{−}(L_{1})h^{−}(L_{2})T_{1}T_{2},

where T_{1} = h^{−}(L_{1}L_{2}^{+})/h^{−}(L_{1}) and T_{2} = h^{−}(L_{2}L_{1}^{+})/h^{−}(L_{2}). If we assume that κ_{1} = κ_{2} = 1 (κ_{1} is the group of ideal classes capitulating in L_{1}L_{2}^{+}/K, κ_{2} is defined similarly) and that

(L_{1}L_{2}^{+} ∩ L_{1}^{1} : L_{1}) = (L_{1}^{+}L_{2}^{+} ∩ (L_{1}^{+})^{1} : L_{1}^{+}),
(L_{2}L_{1}^{+} ∩ L_{2}^{1} : L_{2}) = (L_{2}^{+}L_{1}^{+} ∩ (L_{2}^{+})^{1} : L_{2}^{+}),

then T_{1} and T_{2} are integers.

Proof. Theorem 3 follows directly from Theorem 2 and Proposition 2. □

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Now let m = p^{μ} and n = q^{ν} be prime powers, and suppose that p ≠ q Moreover, let L_{1} ⊆ ℚ(ζ_{m}) and L_{2} ⊆ ℚ(ζ_{n}) be CM-fields. Then

(1) Q(L) = 2, Q_{1} = Q(L_{1}L_{2}^{+}) = Q_{2} = Q(L_{2}L_{1}^{+}) = 1: this has been proved in Prop. 1.f) and Example 3 in Sect. 2;

(2) w_{1}w_{2} = 2w_{L} (obviously);

(3) κ_{1} = κ_{2} = 1: see Example 3 in Sect. 2;

(4) (L_{1}L_{2}^{+} ∩L_{1}^{1} : L_{1}) = (L_{1}^{+}L_{2}^{+} ∩(L_{1}^{+})^{1} : L_{1}^{+}): this, as well as the corresponding property for K_{2}, is obvious, because the prime ideals above p and q ramify completely in L/L_{2} and L/L_{1}, respectively.

In particular, we have the following

Corollary. (Metsänkylä) Let L_{1} ⊆ ℚ(ζ_{m}) and L_{2} ⊆ ℚ(ζ_{n}) be CM-fields, where m = p^{μ} and n = q^{ν} are prime powers, and let L = L_{1}L_{2}; then

h^{−}(L) = h^{−}(L_{1})h^{−}(L_{2})T_{1}T_{2},

where T_{1} = h^{−}(L_{1}L_{2}^{+})/h^{−}(L_{1}) and T_{2} = h^{−}(L_{2}L_{1}^{+})/h^{−}(L_{2}) are integers.

It still remains to identify the character sums T_{01} and T_{10} in [M] with the class number factors T_{1} and T_{2} given above. But this is easy: the character group X(L_{1}) corresponding to the field L_{1} is generated by a character χ_{1}, and it is easily seen that

\begin{aligned}X(L_{1}) & = & ⟨χ_{1}⟩, & X(L_{1}L_{2}^{+}) & = & ⟨χ_{1}, χ_{2}^{2}⟩, \\ X(L_{2}) & = & ⟨χ_{2}⟩, & X(L_{2}L_{1}^{+}) & = & ⟨χ_{2}, χ_{1}^{2}⟩, \\ X(L) & = & ⟨χ_{1}, χ_{2}⟩, & X(L^{+}) & = & ⟨χ_{1}χ_{2}, χ_{1}^{2}⟩. \end{aligned}

The analytical class number formula for an abelian CM-field K reads

h^{−}(K) = Q(K)w_{K} ∏_{χ∈X^{−}(K)} \frac{1}{2𝔣(χ)} ∑_{a mod^{+} 𝔣(χ)}(−χ(a)a),  (3)

where a mod^{+} 𝔣(χ) indicates that the sum is extended over all 1 ≤ a ≤ 𝔣(χ) such that (a, 𝔣(χ)) = 1, and X^{−}(L) = X(L) \ X(L^{+}) is the set of χ ∈ X(L) such that χ(−1) = −1. Applying formula (3) to the CM-fields listed above and noting that Q(L) = 2, Q(L_{1}) = Q(L_{2}) = Q(L_{1}L_{2}^{+}) = Q(L_{2}L_{1}^{+}) = 1, 2w_{L} = w_{1}w_{2}, we find

h^{−}(L) = h^{−}(L_{1}) · h^{−}(L_{2}) ∏_{χ∈X^{∗}(L)} \frac{1}{2𝔣(χ)} ∑_{a mod^{+} 𝔣(χ)}(−χ(a)a),

where X^{∗}(L) is the subset of all χ ∈ X^{−}(L) not lying in X^{−}(L_{1}) or X^{−}(L_{2}). Now define X_{1}(L) = {χ = χ_{1}^{x}χ_{2}^{y} ∈ X^{∗}(L) : x ≡ 1 mod 2, y ≡ 0 mod 2}, and let X_{2}(L) be defined accordingly. Then X^{∗}(L) = X_{1}(L) ∪ X_{2}(L), and

h^{−}(L_{1}) · ∏_{χ∈X_{1}(L)} \frac{1}{2𝔣(χ)} ∑_{a mod^{+} 𝔣(χ)}(−χ(a)a) = h^{−}(L_{1}L_{2}^{+}),

and we have shown that

T_{1} = ∏_{χ∈X_{1}(L)} \frac{1}{2𝔣(χ)} ∑_{a mod^{+} 𝔣(χ)}(−χ(a)a).

Comparing with the definition of Metsänkylä’s factor T_{10}, this shows that indeed T_{1} = T_{10}.

References

[F] B. Ferrero, The cyclotomic ℤ_{2}−extension of imaginary quadratic number fields, American J. Math. 102 (1980), 447–459.

[H] H. Hasse, Über die Klassenzahl abelscher Zahlkörper, Springer Verlag, Berlin, Heidelberg, New York, Tokyo, 1985.

[L] F. Lemmermeyer, Kuroda’s class number formula, Acta Arith. 66 (1994), 245–260.

[Lou] S. Louboutin, Determination of all quaternion octic CM-fields with class number 2 (submitted).

[M] T. Metsänkylä, Über den ersten Faktor der Klassenzahl des Kreiskörpers, Ann. Acad. Sci. Fenn. Ser. A. I 416 (1967).

[MM] J. M. Masley, H. L. Montgomery, Cyclotomic fields with unique factorization, J. Reine Angew. Math. 286/287 (1976), 248–256.

[W] L. Washington, Introduction to Cyclotomic Fields, Springer Verlag, New York, Heidelberg, Berlin, 1982.

[We] J. Westlund, On the class number of the cyclotomic number field, Trans. Amer. Math. Soc. 4 (1903), 201–212.

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