A note on integral bases of unramified cyclic extensions of prime degree, II

Humio Ichimura^{∗}and Hiroki Sumida^{†}

October 30, 2000

Abstract

Let p be a prime number and K a number field containing a primitive p–th root of unity. It is known that an unramified cyclic extension L/K of degree p has a power integral basis if it has a normal integral basis. We show that for all p, the converse is not true in general.

1 Introduction

This is a sequel to the previous papers [10, 11, 12, 13]. For a finite extension L/K of a number field K, it has a power integral basis (PIB for short) when O_{L} = O_{K}[α] for some α ∈ O_{L}. Here, O_{L} (resp. O_{K}) is the ring of integers of L (resp. K). If L/K is Galois, it has a normal integral basis (NIB for short) when O_{L} is free of rank one over the group ring O_{K}[Gal(L/K)]. Let p be a prime number and K a number field containing a primitive p–th root ζ_{p} of unity. Then, it is known that an unramified cyclic extension L/K of degree p has a PIB if it has a NIB (cf. Childs [3], [11]). Here and in what follows,

1Department of Mathematics, Yokohama City University, 22–2, Seto, Kanazawa–ku, Yokohama, 236–0027, Japan. e-mail:ichimura@math.yokohama-cu.ac.jp Partially supported by Grant–in–Aid for Scientific Research (C) (No.11640041), the Ministry of Education, Science, Sports and Culture of Japan.

2Faculty of Integrated Arts and Sciences, Hiroshima University, Kagamiyama, Higashi– Hiroshima, 739–8521, Japan. e-mail:sumida@mis.hiroshima-u.ac.jp Partially supported by Grant–in–Aid for Encouragement of Young Scientists (No.11740020), the Ministry of Education, Science, Sports and Culture of Japan. Mathematics Subject classification (1991):11R33, 11R23.

an extension of a number field is “unramified” when it is unramified at all finite prime divisors. On the other hand, we showed in [10, 12, 13] that when p = 2, 3, there exist infinitely many number fields K with ζ_{p} ∈ K^{×} each of which has an unramified cyclic extension of degree p with PIB but no NIB. The main purpose of this note is to show that this assertion holds for all p. Namely, we prove the following:

Theorem 1. Let p be an odd prime number, and N a multiple of (p−1)p. Then, there exist infinitely many number fields K of degree N each of which contains ζ_{p} and has an unramified cyclic extension of degree p with PIB but no NIB.

In the next section, we give more precise statements after recalling some notation and related assertions.

2 Theorems

Let p be a fixed prime number, K a number field not necessarily containing ζ_{p}, and E = E_{K} the group of units of K. Put π = ζ_{p} − 1. An element α ∈ K^{×} relatively prime to p is “singular primary” when (α) = 𝔄^{p} for some ideal 𝔄 of K and α ≡ u^{p} mod π^{p} for some u ∈ O_{K}. The class in K^{×}/(K^{×})^{p} represented by α is written in the form [α] or [α]_{K}. We define subgroups H(K), E(K), N (K) of K^{×}/(K^{×})^{p} by

H(K) := {[α] ∈ K^{×}/(K^{×})^{p} | α is singular primary},
E(K) := H(K) ∩ E(K^{×})^{p}/(K^{×})^{p},
N (K) := {[ϵ] ∈ E(K^{×})^{p}/(K^{×})^{p} | ϵ ∈ E, ϵ ≡ 1 mod π^{p}}.

Clearly, we have

N (K) ⊆ E(K) ⊆ H(K).

We write (E/N)(K) for the quotient E(K)/N (K). We often regard these groups as vector spaces over F_{p} = Z/pZ.

Let us assume that ζ_{p} ∈ K^{×}. Then, it is well known (cf. Washington [23, Exercises 9.2, 9.3]) that for [α] ∈ K^{×}/(K^{×})^{p}, the cyclic extension K(α^{1/p})/K is unramified if and only if [α] ∈ H(K). In [3], Childs proved that for [α] ∈ H(K), K(α^{1/p})/K has a NIB if and only if [α] ∈ N (K). Further, F. Kawamoto, N. Suwa and the first author independently proved that for [α] ∈ H(K), K(α^{1/p})/K has a PIB if [α] ∈ E(K), for which see [11]. From the above, our target is the quotient group (E/N)(K).

Assume further that K is a CM–field and that p ≥ 3. Then, by the action of the complex conjugation, we can decompose each group defined above into the product of the even part and the odd part:

H(K) = H(K)^{+} ⊕ H(K)^{−}, etc.

Let μ(K) = ⟨ζ_{p^{a}}⟩ be the group of p–power roots of unity in K, where ζ_{p^{a}} is a primitive p^{a}–th root of unity. From the well known theorem on the units of CM–fields (cf. [23, Theorem 4.12]), it immediately follows that

E(K)^{−} ⊆ ⟨[ζ_{p^{a}}]⟩, and hence dim E(K)^{−} ≤ 1,  (1)

where dim(∗) denotes the dimension of a vector space over F_{p}. It also follows from the above mentioned theorem that N (K)^{−} = {0}, for which see also Brinkhuis [1]. Therefore, we can say that the odd part (E/N)(K)^{−} = E(K)^{−} is a “tame” object. On the other hand, the even part (E/N)(K)^{+} is a “tough” object because, to deal with it, we have to fight with the group of units of the maximal real subfield of K. We prove the following theorems (Theorems 2, 3) on the odd part and the even part. Theorem 1 follows immediately from Theorem 2.

Theorem 2. Let p be an odd prime number, and N a multiple of (p−1)p. Then, there exist infinitely many CM–fields K of degree N for which ζ_{p} ∈ K^{×} and (E/N)(K)^{−} ≠ {0}.

Theorem 3. Let p be an odd prime number with p < 100, and N a proper multiple of 2(p − 1)p with N/(2(p − 1)) not a power of p. Then, there exist infinitely many CM–fields K of degree N for which ζ_{p} ∈ K^{×} and (E/N)(K)^{+} ≠ {0}.

This note is organized as follows. In Section 3, we give some simple lemmas on (E/N)(K). In Section 4, we prove Theorem 2. In section 5, we give a sufficient condition for (E/N)(K)^{+} ≠ {0} using some results in cyclotomic Iwasawa theory. In Section 6, we prove Theorem 3.

3 Some lemmas

In this section, we give some simple lemmas on the quotient (E/N)(K). Unless otherwise stated, p is a prime number including p = 2, and K is an arbitrary number field. As before, we denote by μ(K) the group of p–power roots of unity in K.

Lemma 1. (I) Let L/K be a finite extension with p ∤ [L : K]. Then, (E/N)(L) ≠ {0} if (E/N)(K) ≠ {0}. (II) Let L/K be a finite extension with p ∤ [L : K]. Assume that p ≥ 3 and that K, L are CM–fields. Then, (E/N)(L)^{±} ≠ {0} if (E/N)(K)^{±} ≠ {0}.

Proof. We prove only the first assertion. The second one is proved similarly. Let ϵ be a unit of K with [ϵ]_{K} ∈ E(K). Assume that [ϵ]_{L} ∈ N (L). Then, ϵ ≡ η^{p} mod π^{p} for some η ∈ E_{L}. Taking the norm from L to K, we obtain

ϵ^{n} ≡ (N_{L/K} η)^{p} mod π^{p} with n = [L : K].

This implies [ϵ]_{K} ∈ N (K) since p ∤ n. Hence, we obtain the assertion (I). □

Lemma 2. Let K be a number field. If the ramification index over Q of any prime ideal of K dividing p is smaller than p, then (E/N)(K) = {0}. In particular, if [K : Q] < p, then (E/N)(K) = {0}.

Proof. Let ϵ be a unit of K. Assume that ϵ ≡ u^{p} mod π^{p} for some u ∈ O_{K}. Replacing ϵ with ϵ^{n} for some n with p ∤ n, we may well assume that u ≡ 1 mod 𝔓 for all prime ideals 𝔓 of K over p. Then, we must have u^{p} ≡ 1 mod π^{p} since the ramification index of 𝔓 is smaller than p for any 𝔓 with 𝔓|p. Thus, we obtain the assertion. □

Lemma 3. (I) Let K be a number field with μ(K) = ⟨ζ_{p^{a}}⟩ and a ≥ 1, and F the maximal abelian field contained in K. We have [ζ_{p^{a}}] ∉ E(K) if K/F is at most tamely ramified at a prime divisor of K over p. (II) Let p ≥ 3, K a CM–field with ζ_{p} ∈ K^{×}, and F as above. We have E(K)^{−} = {0} if K/F is at most tamely ramified at a prime divisor of K over p.

Proof. The first assertion holds since the extension F (ζ_{p^{a+1}})/F is of degree p and ramified at the primes over p. The second one follows from the first one and (1). □

Remark 1. In the statement of Theorem 2, the degree N = [K : Q] must be a multiple of p because of Lemma 3.

4 Proof of Theorem 2

Let p be an odd prime number, N a multiple of (p − 1)p, and n = N/p. Choose and fix a CM–field k of degree n with μ(k) = ⟨ζ_{p}⟩. Let l be a prime number which splits completely in k, and 𝔏 a prime ideal of k over l. Take an element β of k^{×} satisfying (a) β ≡ 1 mod π^{p} and (b) 𝔏||β but 𝔏^{ρ} ∤ β. Here, ρ denotes the complex conjugation in k. Put α = ζ_{p}β^{1−ρ}, γ = α^{1/p} and K = k(γ). By the condition (b), K/k is a cyclic extension of degree p in which 𝔏 is ramified. Let k^{+} be the maximal subfield of k. Since α^{ρ} = α^{−1}, we see that there exists a cyclic extension K^{+}/k^{+} of degree p such that K = kK^{+} from the Kummer duality. Hence, K is a CM–field of degree N = np. By the condition (a), ζ_{p} ≡ γ^{p} mod π^{p}. It is clear that μ(K) = ⟨ζ_{p}⟩. Therefore, the class [ζ_{p}](∈ K^{×}/(K^{×})^{p}) is a nontrivial element of E(K)^{−}. Thus, we obtain (E/N)(K)^{−} = E(K)^{−} ≠ 0. Finally, by taking various prime numbers l, we complete the proof of Theorem 2.

5 A sufficient condition for (E/N)(K)^{+} ≠ {0}

In this section, we give a sufficient condition for (E/N)(K)^{+} ≠ {0} using some results in cyclotomic Iwasawa theory. Let p be a fixed odd prime number, K an imaginary abelian field and Δ = Gal(K/Q). We assume that K satisfies the condition

(C1) ζ_{p} ∈ K^{×} and the exponent of Δ equals p − 1.

Let K_{∞}/K be the cyclotomic Z_{p}–extension with its n–th layer K_{n} (n ≥ 0). For brevity, we write H_{n}, E_{n}, N_{n} in place of H(K_{n}), E(K_{n}), N (K_{n}) respectively. Let A_{n} be the Sylow p-subgroup of the ideal class group of K_{n}, and let

X_{∞} = lim_{⟵} A_{n}

be the projective limit with respect to the relative norms. These groups are naturally regarded as modules over the group ring Z_{p}[Δ]. By definition, for each [α] = [α]_{n} = [α]_{K_{n}} ∈ H_{n}, there exists an ideal 𝔄 of K_{n} such that (α) = 𝔄^{p}. By mapping [α] to the ideal class [𝔄] ∈ A_{n}, we obtain the following exact sequence of Z_{p}[Δ]–modules.

{0} ⟶ E_{n} ⟶ H_{n} ⟶ A_{n}.  (2)

For a Z_{p}[Δ]–module M and a (Q_{p}–valued) character ψ of Δ, M(ψ) denotes the ψ–component of M. Namely, M(ψ) is the maximal submodule of M on which Δ acts via ψ. We denote by λ_{ψ} and μ_{ψ} the Iwasawa λ–invariant and the μ–invariant of the ideal class group X_{∞}(ψ), respectively. We have μ_{ψ} = 0 by Ferrero and Washington [4]. Let χ be a fixed nontrivial even (Q_{p}– valued) character of Δ, ω the character of Δ representing the Galois action on ζ_{p}, and χ^{∗} = ω · χ^{−1} the associated odd character. By the Iwasawa main conjecture (= the theorem of Mazur and Wiles [20]), we can calculate the invariant λ_{χ^{∗}} using “Stickelberger elements”. And there are several values of λ_{χ^{∗}}, for which see Fukuda’s table [5]. On the other hand, it is conjectured that λ_{χ} = 0 by Greenberg [7]. Though this conjecture is far from being settled, a method to calculate λ_{χ} is established by Kraft and Schoof [18], Kurihara [19] and the authors [14, 15].

Under the above setting, we assume that K and χ satisfy the following two conditions.

(C2) λ_{χ^{∗}} = 1 and λ_{χ} = 0.

(C3) There is only one prime ideal of K over p.

As μ_{χ^{∗}} = 0, it follows from λ_{χ^{∗}} = 1 and (C3) that

H_{n}(χ) ≅ Z/pZ for all n ≥ 0  (3)

using the Kummer duality. For this assertion, see for example, Section 5.1 of [9]. Let [α_{0}]_{0} be a generator of H_{0}(χ) with α_{0} ∈ K^{×}, and 𝔄_{0} an ideal of K such that (α_{0}) = 𝔄_{0}^{p}. Assume further that

(C4) E_{0}(χ) = {0}.

Then, by (the χ–component of) the exact sequence (2) and (3) with n = 0, we see that 𝔄_{0} is not a principal ideal of K (and hence, A_{0}(χ) ≠ {0}). Since λ_{χ} = 0, the ideal 𝔄_{0} is capitulated in K_{n} for some n by [7, Proposition 2]. Denote by m_{0} the smallest such integer.

Theorem 4. Under the above setting, assume that K and χ satisfy the four conditions (C1),⋯, (C4). Then, H_{n}(χ) = E_{n}(χ) for all n ≥ m_{0}, and N_{n}(χ) = {0} for all n ≥ 0. In particular, E_{n}(χ)/N_{n}(χ) ≠ {0} for all n ≥ m_{0}.

Proof. By (3), H_{n}(χ) is generated by the class [α_{0}]_{n}. Then, since 𝔄_{0} is a principal ideal in K_{n} for n ≥ m_{0}, the first assertion follows from (the χ–component of) the exact sequence (2). By (C4), N_{0}(χ) = {0}. Because of the conditions (C1), (C2), (C3), this implies that N_{n}(χ) = {0} for all n ≥ 0 by virtue of [9, Proposition 1]. □

6 Proof of Theorem 3

For proving Theorem 3, it suffices to show the following proposition because of Lemma 1 (II).

Proposition. Let p be an odd prime number with p < 100, and a (≥ 1) an integer. Then, there exists at least one imaginary abelian field K of degree 2(p − 1)p^{a} for which ζ_{p} ∈ K^{×} and (E/N)(K)^{+} ≠ {0}.

Let k = Q(\sqrt{f}) be a real quadratic field with conductor f, χ the associated even Dirichlet character, and K = k(ζ_{p}). We regard χ as a character of Δ = Gal(K/Q). Clearly, K satisfies (C1). In view of Theorem 4, it suffices, for showing Proposition 2, to give at least one numerical example of k for which K and χ satisfy the conditions (C2), (C3), (C4) and m_{0} = 1. The examples are exhibited in the tables at the end of this note for p < 100. To check whether a given pair (K, χ) satisfies the above conditions, the hardest part is to verify λ_{χ} = 0 and m_{0} = 1. We briefly explain how to verify them following [15], after mentioning some simple remarks. Further, we explain how to look at the tables.

It is clear that (C4) is equivalent to E(k) = {0}. Let ϵ be a fundamental unit of k. Then, the condition E(k) = {0} holds if and only if

ϵ^{p^{2}−1} ≢ 1 mod p^{2} or ϵ^{p−1} ≢ 1 mod p(p, \sqrt{f})

according as p ∤ f or p|f. This is shown by an argument similar to the proof of Lemma 2. Hence, the condition (C4) is quite easily checked. As we have mentioned in Section 5, we have

A_{0}(χ) ≠ {0}  (4)

when λ_{χ^{∗}} = 1 and (C3), (C4) hold.

Let q be the least common multiple of f and p. By Iwasawa [17], there exists a unique power series g_{χ}(T) in Z_{p}[[T]] related to the p–adic L–function L_{p}(s, χ) by

g_{χ}((1 + q)^{1−s} − 1) = L_{p}(s, χ), for all s ∈ Z_{p}.

When λ_{χ^{∗}} = 1, g_{χ}(T) has a unique zero α (∈ pZ_{p}). We have α ≠ 0 because the Leopoldt conjecture holds for K by Brumer [2]. We can calculate the value α mod p^{n} using the approximation formula [17, Section 6] for g_{χ}(T).

For a while, we assume that the conditions (C3), (4) and λ_{χ^{∗}} = 1 are satisfied. In [15], we introduced, for each n ≥ 0, a condition (H_{n}) which is given in terms of an explicitly written cyclotomic unit of K_{n} and the value α mod p^{n+e}, where e = ord_{p}α. The main theorem in [15] asserts that λ_{χ} = 0 if and only if (H_{n}) holds for some n ≥ 0. Let f′ be the non–p–part of f. For each prime number ℓ with ℓ ≡ 1 mod f′p^{n+e}, we introduced a condition (H′_{n,ℓ}) which is a kind of “reduction modulo ℓ” of (H_{n}) and for which it is quite easy to check whether or not hold by computer calculation. We showed that (H_{n}) holds if and only if (H′_{n,ℓ}) holds for some ℓ ([15, Proposition 2]). Let n_{0} be the smallest integer n for which k satisfies (H_{n}). We also showed that m_{0} = 1 if n_{0} = 0 and that m_{0} = n_{0} otherwise. (For this, see [15, Proposition 1].) Further, it is known (that |A_{0}(χ)| ≤ p^{e} and) that (H_{0}) does not hold if and only if

|A_{0}(χ)| = p^{e}  (5)

holds by [15, Remark 4]. The last condition (or equivalently, the opposite of (H_{0})) is equivalent to (C4) except when p = 3 and p ramifies in k. For the exceptional case, (C4) implies (5). For these, see Section 5.3 of [9]. From the above, excluding the exceptional case, a real quadratic field k = Q(\sqrt{f}) (or the associated pair (K, χ)) satisfies the conditions (C1),⋯, (C4) and m_{0} = 1 if it satisfies (C3), (4), λ_{χ^{∗}} = 1 and n_{0} = 1.

Our computer calculation was performed as follows. First, we check whether or not (H_{0}) is satisfied using (5). When (H_{0}) does not hold, we check, one by one, starting from n = 1, whether or not (H′_{n,ℓ}) is satisfied for the first five prime numbers ℓ with ℓ ≡ 1 mod f′p^{n+e} until it is satisfied for some n and some such l. When (H_{0}) does not hold and (H′_{1,l}) is satisfied for some such l, we can conclude that n_{0} = 1. However, in general, the method is not efficient at calculating the exact value of n_{0}, and it gives an upper bound of n_{0}. This is because we deal only with relatively small primes l for each n.

Table I deals with prime numbers p with p ≥ 11 and all real quadratic fields k = Q(\sqrt{f}) with f < 100, 000 satisfying (C3) and (4). For p = 31, 41 and p > 47, there are no such fields in the range f < 100, 000. Corresponding tables for p = 3, 5, 7 are given in [15]. For each f in the row “n_{0} = i” (i = 0, 1, 2), we have λ_{χ^{∗}} = 1. For each f in the row “n_{0} = 0” (resp. “n_{0} = 1”), we showed that n_{0} = 0 (resp. n_{0} = 1) by the above method. For each f in the row “n_{0} = 2”, we have n_{0} ≤ 2. For these f, we saw that k = Q(\sqrt{f}) does not satisfy (H_{0}) nor (H′_{1,l}) for the first five prime numbers l with l ≡ 1 mod f′p^{1+e} but it satisfies (H′_{2,l}) for some of the first five primes l with l ≡ 1 mod f′p^{2+e}. As we explained above, we can not exclude the possibility of n_{0} = 1 for these f. (For this, see Remark 2.) For each f in the row “@”, we have λ_{χ^{∗}} > 1 and we have verified λ_{χ} = 0 by the method in [14]. Further, the ∗–mark after the value f means that p ramifies in k. For the other f, p remains prime in k.

For each prime number p with 3 ≤ p < 100, Table II gives a list of the smallest f for which k = Q(\sqrt{f}) satisfies the conditions λ_{χ}^{∗} = 1, (C3), (C4), (4), and m_{0} = 1. (We showed that for f_{1} with f_{1} < f, k = Q(\sqrt{f_{1}}) does not satisfy (C4) when it satisfies (C3), (4) and λ_{χ^{∗}} = 1.) We also give, for each such f, the value of α mod p^{2} and the smallest prime ℓ with ℓ ≡ 1 mod f′p^{1+e} for which (H′_{1,ℓ}) holds.

Remark 2. Recently, in [21], the second author exploited a method to calculate the exact value of n_{0} using not only cyclotomic units but also Gauss sums.

Remark 3. As we have mentioned in Section 2, the difficulty in proving (E/N)(K)^{+} ≠ {0} lies in that we need a knowledge on the p–adic behaviour of the full group E_{K} of all units. For abelian fields, we have a beautiful theorem of Iwasawa [16] and Gillard [6] on local units modulo cyclotomic units. Under some conditions, we can use this for obtaining some ample information on E_{K} for abelian fields K. Proposition 1 of [9] which is crucial in the proof of Theorem 4 was proved in this way. Greither [8] and, recently, Tsuji [22] gave some generalization of this important theorem of Iwasawa and Gillard.

TABLE I

p = 11

f
n_{0} = 0 36709 51553 91585

n_{0} = 1 10401 14009 19021 19048 20369 22129 22501 24801 27473 32236 33833 43753 49953 50937 51457 51985 53349 55281 55336 55948 57409* 57713 65361 65797 67341 69209 69729* 78889 83569 84685 86017 86869 91384 91913 92265 92408 95477 97576

n_{0} = 2 37353 65353
@  1297 12161 26617 74857 91769

p = 13

f
n_{0} = 0 13033

n_{0} = 1 8101 13457 14113 15377 18817 20977 21613 31241 33209 33857 34588 35297 39193 39201 40669 55569 58661 60029 61033 64313 68881 69009 77149 78028 79633 81785 83969 85265 90040 90313 90329 92417* 97973

n_{0} = 2 24601 31193 40441 41801 45329 61989
@  26241 82373 83377

p = 17

f
n_{0} = 1 11257 42937 47657 54541 55697 63505 65473 69697 79009

p = 19

f
n_{0} = 1 31333 38569 44101 49393 54753 68281 70429 71689 97141
@  18229 39801

p = 23

f
n_{0} = 1 30977 56065 67409 91813

p = 29

f
n_{0} = 1 49281 56857 90001 99401

p = 37

f
n_{0} = 1 55561 94321

p = 43

f
n_{0} = 0 14401

p = 47

f
n_{0} = 1 78401

TABLE II

\begin{matrix}p & f & α mod p^{2} & l \\ 3 & 761 & 3 & 27397 \\ 5 & 1093 & 15 & 437201 \\ 7 & 577 & 35 & 113093 \\ 11 & 10401 & 33 & 7551127 \\ 13 & 8101 & 156 & 16428829 \\ 17 & 11257 & 170 & 39039277 \\ 19 & 31333 & 171 & 248846687 \\ 23 & 30977 & 230 & 196641997 \\ 29 & 49281 & 812 & 414453211 \\ 31 & 158649 & 372 & 2744310403 \\ 37 & 55561 & 740 & 3042520372 \\ 41 & 205753 & 943 & 1383483173 \\ 43 & 189229 & 817 & 41986130531 \\ 47 & 78401 & 2021 & 34637561811 \\ 53 & 312361 & 1643 & 10529064589 \\ 59 & 360697 & 2124 & 87891037991 \\ 61 & 586321 & 3233 & 61087612349 \\ 67 & 614657 & 67 & 38628733823 \\ 71 & 812449 & 3692 & 40955554091 \\ 73 & 444089 & 4745 & 255587430349 \\ 79 & 641521 & 3397 & 160149302441 \\ 83 & 1022869 & 4067 & 211396336231 \\ 89 & 614849 & 7031 & 68183065007 \\ 97 & 1106209 & 1164 & 603682587899 \end{matrix}

Acknowledgements. We are grateful to the referee for carefully reading the manuscript and for valuable comments. In Theorems 1 and 2 of the former version, we had imposed the stronger condition p^{2}|N for a technical reason. The referee kindly informed us how to simplify the original proof of Theorem 2 replacing the condition p^{2}|N with p|N.

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