Construction of Bases for the Group of Cyclotomic Units

Marc Conrad Fachbereich Mathematik Universität des Saarlandes Postfach 15 11 50 D–66041 Saarbrücken Germany

1998

Abstract

Subject of investigation is the construction of a basis B_{n} for the group of cyclotomic units of the nth cyclotomic field. These bases have the property that B_{d} ⊆ B_{n} for d|n.

For this purpose the notion of weak σ-bases of a Z-module with an involution operating on it is introduced. The weak σ-basis of a special module then leads via explicit given isomorphisms to the desired bases for the group of cyclotomic units.

1 Introduction

For n ∈ N let ϵ_{n} be a primitive nth root of unity and D^{(n)} the multiplicative group generated by the elements 1 − ϵ_{n}^{k} with k ≢ 0 mod n modulo roots of unity in order to avoid torsion. The group of cyclotomic units C^{(n)} is defined as the subgroup of D^{(n)} containing the elements which are units in Z[ϵ_{n}]. This group has been often subject of investigation ([2], [3], [6], [7], [8], …) and is well known in the case that n is the power of a prime.

In the past, Ramachandra [5] found a system of independent units in C^{(n)} for general n generating a subgroup of finite index. Later, Kučera [4] constructed a basis for C^{(n)} using the notion of distributions. Here, a different approach is used which leads to a basis for C^{(n)} and C^{(∞)} = ⋃_{n∈N} C^{(n)}. We consider free Z-modules M with an involution σ operating on it and show different methods to construct bases for M_{+} = M/ ker_{M}(1 + σ). We further introduce special systems of modules which are defined as a set of triples (M_{d}, E_{d}, n_{d})_{d∈Δ} where Δ is an ordered indexing set, the M_{d} are modules, E_{d} ⊆ M_{d} and n_{d} : E_{d} → ⨁_{t<d} M_{t} is a mapping for each d ∈ Δ. Explicit conditions are given how to handle the combination

L = (⨁_{d∈Δ} M_{d})/ ∑_{d∈Δ}⟨r + n_{d}(r); r ∈ E_{d}⟩  (1)

of the system, and we work out how to construct a basis for L_{+} using special bases, the so called weak σ-bases of the modules M_{d}/⟨E_{d}⟩. For well chosen input parameters Δ, M_{d}, E_{d} and n_{d} we obtain as combination a module L(n) for which an isomorphism L(n)_{+} ≅ D^{(n)} holds. This leads via weak σ-bases of the modules M_{d}/⟨E_{d}⟩ to a basis for D^{(n)} which can be easily modified to a basis for the group of cyclotomic units C^{(n)}. By this construction, we obtain a basis B_{n} for C^{(n)} such that B_{d} ⊆ B_{n} whenever d|n. This leads obviously to a basis for C^{(∞)}. In the last section there is a short discussion about further applications of these methods. These are the explicit construction of the relations in C^{(n)} and similar results for the Stickelberger ideal as for cyclotomic units.

2 Weak σ-bases

In the following every module is a Z-module. When we speak of a basis, the rank or that the module is free we mean a Z-basis, the Z-rank or Z-free. Unless not stated otherwise the modules are finitely generated. We assume further that an involution σ operates on each module. The homomorphisms between two modules will be always compatible with the action of σ. So we will have in fact Z[σ]-modules and Z[σ]-homomorphisms.

For a set B, a module M and a mapping ξ : B → M we say that B induces a basis of M if the set {ξ(b); b ∈ B} is a basis of M. Saying that some set is a basis of M includes that M is free. The involution σ then defines the two modules M_{−} = M/ ker_{M}(1 − σ) and M_{+} = M/ ker_{M}(1 + σ). A weak σ-basis of a module M is defined as a triple [E^{0}, E^{+}, E^{−}] of subsets of M such that the union B = E^{0} ∪ σE^{0} ∪ E^{+} ∪ E^{−} is disjoint, B is a basis of M and the two conditions

(i) σe ≡ e mod ⟨E^{0} ∪ σE^{0}⟩ for e ∈ E^{+}

(ii) σe ≡ −e mod ⟨E^{0} ∪ σE^{0}⟩ for e ∈ E^{−}

hold. We write B = [E^{0}, E^{+}, E^{−}] for short regarding the triple as a subset of M. Note that according to [1] we have a σ-basis if we replace the equivalence modulo ⟨E^{0} ∪ σE^{0}⟩ by equality. It can be shown that every module has a σ-basis. We state some results about weak σ-bases which can be straightforwardly proved. For details see [1].

(I) E^{0} ∪ E^{+} induces a basis of M_{+} and E^{0} ∪ E^{−} induces a basis of M_{−}.

(II) The values m^{+} = m^{+}(M) = |E^{+}| and m^{−} = m^{−}(M) = |E^{−}| are invariants of M. They are independent of a special choice of E^{+} or E^{−}. More concretely, m^{+} is the dimension of the F_{2}-vector space H^{0}(σ, M), and m^{−} is the dimension of H^{1}(σ, M).

(III) Let C = [F^{0}, F^{+}, F^{−}] a weak σ-basis of a second module L. Then [G^{0}, G^{+}, G^{−}] ⊆ M × L with

G^{0} = (E^{0} × C) ∪ (E^{+} × F^{0}) ∪ (E^{−} × F^{0}), G^{+} = (E^{+} × F^{+}) ∪ (E^{−} × F^{−}), G^{−} = (E^{+} × F^{−}) ∪ (E^{−} × F^{+})

induces a weak σ-basis of M ⊗ L.

(IV) Let x be one of the symbols + or −. Given an exact sequence of free

modules

0 → M → L → K → 0  (2)

the following three statements are equivalent:

(i) The sequence (2) splits over σ.

(ii) The sequences 0 → M_{+} → L_{+} → K_{+} → 0 and 0 → M_{−} → L_{−} →

K_{−} → 0 are exact. (iii) The equation m^{x}(L) = m^{x}(M) + m^{x}(K) holds.

Moreover, if (i)-(iii) hold and [F^{0}, F^{+}, F^{−}] ⊆ L induces a weak σ-basis of K we obtain from (I) and by the commutative diagram

\begin{aligned}0 & → & M & → & L & → & K & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & M_{x} & → & L_{x} & → & K_{x} & → & 0, \end{aligned}  (3)

that E^{0} ∪ E^{x} ∪ F^{0} ∪ F^{x} induces a basis of L_{x}.

Note that if (iii) holds for one value of x then it holds for both.

Repeated application of (IV) leads to the following lemma.

Lemma 2.1 Let x ∈ {+, −} and 0 = L^{(0)} ≤ L^{(1)} ≤ ⋯ ≤ L^{(i)} ≤ ⋯ ≤ L an ascending chain of modules with L = ⋃_{i=0}^{∞} L^{(i)} where L has not to be finitely generated. Suppose that for every i ∈ N there exists a module M^{(i)} such that the sequence

0 → L^{(i−1)} → L^{(i)} → M^{(i)} → 0  (4)

is exact and splits over σ. If B_{x}^{(i)} ⊆ L^{(i)} induces a basis of M_{x}^{(i)} for all i ∈ N then ⋃_{i=1}^{∞} B_{x}^{(i)} induces a basis of L_{x}.

For the rest of this section let Δ be a finite or countable, partially ordered indexing set whose ordering can be completed. We assume further that {d ∈ Δ; d ≤ t} is finite for every t ∈ Δ. Examples are the set of all divisors of a number n ∈ N or N itself ordered by divisibility.

Definition 2.2 For every d ∈ Δ let M_{d} be a module, E_{d} a subset of M_{d} with σE_{d} = E_{d} and n_{d} : E_{d} → ⨁_{t<d} M_{t} a mapping. We call a system of triples (M_{d}, E_{d}, n_{d})_{d∈Δ} a MEn-system. Let N′_{d} = ⨁_{t<d} M_{t} and Q′_{d} = ∑_{t<d}⟨r + n_{t}(r); r ∈ E_{t}⟩ ⊆ N′_{d}. We call the MEn-system Γ combinable, if the mappings n_{d} can be extended to homomorphisms

n̄_{d} : ⟨E_{d}⟩ → N′_{d}/Q′_{d}.  (5)

In this case Γ defines the module L = N/Q with N = ⨁_{t∈Δ} M_{t} and Q = ∑_{t∈Δ}⟨r + n_{t}(r); r ∈ E_{t}⟩. We call L the combination of Γ. The module L has not to be finitely generated.

Definition 2.3 In the case that Δ is finite we say that Γ splits over σ if m^{+}(L) = ∑_{d∈Δ} m^{+}(M_{d}/⟨E_{d}⟩). If Δ is infinite we say that Γ splits over σ if (M_{d}, E_{d}, n_{d})_{d≤t} splits for every t ∈ Δ.

Theorem 2.4 If M_{d}/⟨E_{d}⟩ is free for all d ∈ Δ then the combination L is also free. Under the additional assumption that Γ splits over σ we have for x ∈ {+, −} that if B_{x}^{(d)} ⊆ M_{d} induces a basis of (M_{d}/⟨E_{d}⟩)_{x} for each d ∈ Δ then ⋃_{d∈Δ} B_{x}^{(d)} ⊆ ⨁_{d∈Δ} M_{d} induces a basis of L_{x}.

Proof: We complete the ordering of Δ and therefore assume that Δ = N or Δ = {1, …, n} with the canonical ordering. For i ∈ Δ ∪ {0} let N_{i} = M_{1} ⊕ ⋯ ⊕ M_{i} and

Q_{i} = ∑_{j=1}^{i}⟨r + n_{j}(r); r ∈ E_{j}⟩ ≤ N_{i}.  (6)

We show first Q_{i} ∩ N_{i−1} = Q_{i−1}. We have q ≡ ∑_{e∈E_{i}} α_{e}(e + n_{i}(e)) mod Q_{i−1} with α_{e} ∈ Z for q ∈ Q_{i} Because n̄_{i} is a homomorphism modulo Q_{i−1} it follows

q ≡ r + n̄_{i}(r) mod Q_{i−1}  (7)

with r = ∑_{e∈E_{i}} α_{e}e. Reducing (7) modulo N_{i−1} and assuming q ∈ N_{i−1} we get r = 0. This implies n̄_{i}(r) = 0 in (7) which leads to q ∈ Q_{i−1}.

From Q_{i} ∩ N_{i−1} = Q_{i−1} we deduce inductively Q ∩ N_{i−1} = Q_{i−1} where Q = ⋃_{i∈Δ} Q_{i}. So we have an ascending chain of modules

0 = N_{0}/Q_{0} ≤ N_{1}/Q_{1} ≤ N_{2}/Q_{2} ≤ ⋯ ≤ N/Q  (8)

with exact sequences

0 → N_{i−1}/Q_{i−1} → N_{i}/Q_{i} → M_{i}/⟨E_{i}⟩ → 0.  (9)

The claim of the theorem follows by Lemma 2.1 if we prove that (9) splits over σ. Let x_{i} = m^{+}(N_{i}/Q_{i}) and y_{i} = m^{+}(M_{i}/⟨E_{i}⟩). For all exact sequences we have x_{i} ≤ x_{i−1} + y_{i} and the assumption that Γ splits gives (in the case that n is finite) ∑_{i=1}^{n} x_{i} = y_{n}. Putting this together we obtain x_{i} = x_{i−1} +y_{i} which implies by (IV) that (9) splits over σ. The infinite case can be reduced directly to the finite case. □

Remark 2.5 From the exact sequences (9) we see by induction that a basis of L is given as the union of bases of the modules M_{d}/⟨E_{d}⟩. So Theorem 2.4 is also valid without the “x”.

In the case that Γ does not split over σ it is possible to save the construction by introducing the notion of a derived MEn-system.

Definition 2.6 Assume that 0 is an element which is not in Δ. Let the ordering on Δ ∪ {0} be defined by saying that 0 is smaller than any d ∈ Δ. Then we set Δ^{D} = {0} ∪ {t ∈ Δ; t ≰ D} for D ∈ Δ and define the derived MEn-system Γ^{(D)} = (M_{d}, E_{d}, n_{d})_{d∈Δ^{D}} by setting M_{0} = ⨁_{t≤D} M_{t}, E_{0} = ⋃_{t≤D}{r + n_{t}(r); r ∈ E_{t}} and n_{0} ≡ 0.

With this definition we see directly:

Lemma 2.7 If Γ is combinable then Γ^{(D)} is combinable and the combinations of Γ and Γ^{(D)} are identical.

3 The cyclotomic module

For a finite subset S of a module M we write ΣS for ∑_{s∈S} s. Further, let G_{d} = {1 ≤ a < d; (a, d) = 1}. Especially we have G_{1} = ∅.

Definition 3.1 For n > 1 we define the cyclotomic module Z(n) as follows:

If n = p prime then Z(p) = ⟨G_{p}⟩/⟨ΣG_{p}⟩.

If n = q = p^{α} with α > 1 then Z(q) = ⟨G_{q/p}⟩ ⊗ ⟨A_{p}⟩/⟨ΣA_{p}⟩ with A_{p} = {0, …, p − 1}.

If n = q_{1} ⋯ q_{r} where q_{i} are pairwise relatively prime prime powers then Z(n) = Z(q_{1}) ⊗ ⋯ ⊗ Z(q_{r}).

We define the operation of σ on b ∈ G_{d} by σb = d − b and on a ∈ A_{p} by σa = p − 1 − a. By this Z(n) becomes a module with an involution σ.

Table 1: Weak σ-bases of Z(q)

module  weak σ-basis
Z(2)  [∅, ∅, ∅]
Z(p), p ≠ 2, p prime [{2, …, (p − 1)/2}, ∅, {1}]
Z(4)  [∅, ∅, {(1, 0)}]
{Z(q), q = p^{α}, α > 1, \atop p prime, q ≠ 4} [\{(b, a); 1 ≤ b < \frac{1}{2}q/p, p|̸b, 1 ≤ a < p\}, ∅, ∅]

With (III) from the previous section it is possible to construct weak σ-bases of Z(n) by weak σ-bases of Z(q). Table 1 gives a weak σ-basis for Z(q). Especially, defining g(n) = ∏_{p|n}(φ(p^{α_{p}}) − φ(p^{α_{p}−1})) where α_{p} is maximal with p^{α_{p}}|n and φ is Euler’s function we have rank(Z(n)) = g(n).

Lemma 3.2 There is an isomorphism Z(n) ≅ ⟨G_{n}⟩/R_{n}. The submodule R_{n} of ⟨G_{n}⟩ is generated by E_{n} = {s(n, p, a); p|n with p prime, a ∈ G_{n/p}} with

s(n, p, a) = Σ{x ∈ G_{n}; x ≡ a mod (n/p)}.  (10)

Proof: We arrange the prime factors p_{i} of n such that

n = p_{1}^{α_{1}} ⋯ p_{t}^{α_{t}}p_{t+1} ⋯ p_{r}  (11)

with α_{i} > 1 for i = 1, …, t. Further we set q_{i} = p_{i}^{α_{i}} if i = 1, …, t and q_{i} = p_{i} else. Let

S = G_{q_{1}/p_{1}} × A_{p_{1}} × ⋯ × G_{q_{t}/p_{t}} × A_{p_{t}} × G_{p_{t+1}} × ⋯ × G_{p_{r}}. (12)

It is obvious by the definition of the tensor product that Z(n) is isomorphic to ⟨S⟩ modulo suitable relations. We show that S is as set isomorphic to G_{n}. Using the bijection between S and G_{n} it can be shown directly that the sums of (10) correspond one to one to the sums used in Definition 3.1. We obtain S ≅ G_{n} by combining isomorphisms ξ_{i} and η in the following way:

\begin{aligned}\underbrace{G_{q_{1}/p_{1}} × A_{p_{1}}} & × ⋯ × & \underbrace{G_{q_{t}/p_{t}} × A_{p_{t}}} & × & G_{p_{t+1}} & × ⋯ × & G_{p_{r}} \\ ↓ ξ_{1} & & ↓ ξ_{t} & & ↓ id & & ↓ id \\ G_{q_{1}} & × ⋯ × & G_{q_{t}} & × & G_{q_{t+1}} & × ⋯ × & G_{q_{r}} \\ & & & ↓ η \\ & & & G_{n} \end{aligned} (13)

The maps ξ_{i} for i = 1, …, t are explicitly given by ξ_{i}(b, a) = ap_{i}^{α_{i}−1} +b. The map η is defined by the Chinese remainder theorem, that means η^{−1} is the map a ↦ a mod q_{i} in each component G_{q_{i}}, i = 1, …, r. □ We observe that a weak σ-basis of Z(n) can be constructed with the bases of Table 1 and (III) as a subset of S. We give an example how such a basis looks like. A complete list can be found in [1]. In the case that n is odd and square free [F^{0}, F^{+}, F^{−}] is a weak σ-basis where

F^{0} = ⋃_{i=1}^{r}{(1, …, 1, a_{i}, …, a_{r}) ∈ G_{p_{1}} × ⋯ × G_{p_{r}}; 1 < a_{i} < p_{i}/2
and 1 ≤ a_{j} < p_{j} − 1 for j = i + 1, …, r},
F^{+} = {(1, …, 1)} for r even and F^{+} = ∅ for r odd,
F^{−} = {(1, …, 1)} \ F^{+}.

In the sequel we write N_{∞} = N ∪ {∞} and agree that the set of all d with d|∞ means N (and not N_{∞}).

Definition 3.3 For d ∈ N let M_{d} = ⟨G_{d}⟩. If d is not a prime we denote by E_{d} ⊆ M_{d} the set of the sums s(d, p, a) as in Lemma 3.2. For d prime and d = 1 we define E_{d} = ∅. On E_{d} we define the mapping

n_{d} : E_{d} → ⨁_{t|d,t≠d}M_{t}, s(d, p, a) ↦ \{ \begin{matrix}−[d/p; a] & if p^{2}|d, \\ [d/p; p^{−1}a] − [d/p; a] & if p^{2}|̸d \end{matrix}  (14)

where [m; x] denotes y ∈ G_{m} with x ≡ y mod m. For n ∈ N_{∞} we call the MEn-system Γ(n) = (M_{d}, E_{d}, n_{d})_{d|n} the nth cyclotomic system.

Lemma 3.4 Let n ∈ N_{∞}. Then Γ(n) is combinable.

Proof: We have to show that it is possible to extend each n_{d} to a homomorphism modulo Q′_{d} = ∑_{t|d,t≠d}⟨r + n_{t}(r); r ∈ E_{t}⟩. In order to do so we have to investigate the relations between the elements of E_{d}. The module Z(n) is the tensor product of modules ⟨A⟩/R with suitable sets A where either R is equal zero or is generated by ΣA. In the special case Z(n) ≅ (M_{1}/R_{1}) ⊗ (M_{2}/R_{2}) where R_{1} and R_{2} are generated by the sum of the free generators of M_{1} and M_{2} respectively, the isomorphism (M_{1}/R_{1}) ⊗ (M_{2}/R_{2}) ≅ (M_{1} ⊗ M_{2})/(R_{1} ⊗ M_{2} + M_{1} ⊗ R_{2}) and (R_{1} ⊗ M_{2}) ∩ (M_{1} ⊗ R_{2}) = R_{1} ⊗ R_{2} show that the only relation between the sums is given by the double sum over the generators of M_{1} and M_{2}. By induction and applying the isomorphism of Lemma 3.2 we obtain in the general case that all relations are given by

∑_{{x∈G_{d} \atop x≡a mod d/p′}} s(d, p, x) = ∑_{{x∈G_{d} \atop x≡a mod d/p}} s(d, p′, x)  (15)

where p and p′ are two different primes dividing d. So we have to check

∑_{{x∈G_{d} \atop x≡a mod d/p′}} n_{d}(s(d, p, x)) ≡ ∑_{{x∈G_{d} \atop x≡a mod d/p}} n_{d}(s(d, p′, x)) mod Q′_{d}  (16)

which is done by a straightforward calculation. □ In the sequel we denote by L(n) the combination of Γ(n). For finite n the module L(n) is isomorphic to a module introduced in [6]. We shall use this isomorphism to determine the σ-cohomology of L(n).

Lemma 3.5 Let n > 2 and r be the number of prime factors of n. Then we have:

(i) H^{0}(σ, L(n)) ≅ \{ \begin{matrix}F_{2}^{2^{r−1}−1} & if n ≢ 2 mod 4, \\ F_{2}^{2^{r−2}} & if n ≡ 2 mod 4. \end{matrix}

(ii) H^{1}(σ, L(n)) ≅ \{ \begin{matrix}F_{2}^{2^{r−1}−r} & if n ≢ 2 mod 4, \\ F_{2}^{2^{r−2}−r+1} & if n ≡ 2 mod 4. \end{matrix}

Proof: A similar claim is given in [6], Theorem 2 for a module A = V/U where V is the free module over {e(1), …, e(n − 1)} and U is generated by elements

a(d, x) = e(dx) − ∑_{ν=0}^{d−1} e(x + ν \frac{n}{d})  (17)

with d|n and x = 1, …, n/d − 1. We show A ≅ L(n). We have L(n) = N/Q where N = ⨁_{d|n} M_{d} and Q = ∑_{d|n}⟨r+n_{d}(r); r ∈ E_{d}⟩. The bijection G_{d} ∋ a ↦ e(an/d) shows N ≅ V. With s(d, p, b) as in (10) the isomorphism Q ≅ U is verified by showing that s(d, p, b) + n_{d}(s(d, p, b)) maps to −a(p, bn/d) and noting that the elements a(p, x) where p is prime generate already U. This follows from the relation

a(dt, x) = a(d, tx) + ∑_{j=0}^{d−1} a(t, x + j \frac{n}{dt}),  (18)

for dt|n, d > 1, t > 1 and 1 ≤ x < \frac{n}{dt}. □

Lemma 3.6 Let n ∈ N_{∞}.

(a) Γ(n) splits over σ if and only if n ≢ 0 mod 4.

(b) For n ≡ 0 mod 4 the system Γ(n)^{(4)} splits over σ.

Proof: The system Γ(n) is defined as a MEn-system (M_{d}, E_{d}, n_{d})_{d|n} and we set Y_{d} = M_{d}/⟨E_{d}⟩. So we have

Y_{d} = \{ \begin{matrix}0 & if d = 1, \\ ⟨G_{d}⟩ & if d is prime, \\ Z(d) & otherwise. \end{matrix}  (19)

We have L(2) ≅ Y_{2} ≅ ⟨G_{2}⟩ and L(1) = Y_{1} = 0 so that Lemma 3.6 is valid also in these cases and we assume n > 2 for the rest of the proof. By elementary computations in the first two cases in (19) and from Definition 3.1 in the third case we calculate the rank and the invariants m^{x} of the Y_{d} which are summarized in Table 2 where r(d) denotes the number of distinct primes dividing d.

Table 2: Invariants of Y_{d}

m^{+} m^{−} rank
d = 1  0  0 g(d) − 1
d = 2  1  0 g(d) + 1
d = p an odd prime  0  0 g(d) + 1

d = u or d = 4u where u is odd,

square free and 2|r(d)

1  0  g(d)

d not prime, d = u or d = 4u where u is odd, square free and 2|̸r(d)

0  1  g(d)
all other d  0  0  g(d)

Therefore we have

∑_{d|n} m^{+}(Y_{d}) = \{ \begin{matrix}2^{r−1} − 1 & if n ≡ 1, 3 mod 4, \\ 2^{r−1} & if n ≡ 0 mod 4, \\ 2^{r−2} & if n ≡ 2 mod 4. \end{matrix}  (20)

with r = r(n) the number of distinct primes dividing n. The number m^{+}(L(n)) is the F_{2}-dimension of H^{0}(σ, L(n)). So by Lemma 3.5 we find m^{+}(L(n)) = ∑_{d|n} m^{+}(Y_{d}) if and only if n ≢ 0 mod 4 and part (a) of Lemma 3.6 follows. In the case n ≡ 0 mod 4 and n ≠ ∞ we have m^{+}(L(n)) + 1 = ∑_{d|n} m^{+}(Y_{d}). By definition, Γ(n)^{(4)} is formed by replacing the modules Y_{2} and Y_{4} with L(4). Part (b) follows with m^{+}(L(4)) = m^{+}(Y_{4}) = 0 and m^{+}(Y_{2}) = 1. The case n = ∞ follows directly from the finite case. □

4 Cyclotomic Numbers

We denote as in the introduction by D^{(n)} the group of cyclotomic numbers. Especially we write D^{(∞)} = ⋃_{d∈N} D^{(n)}.

Lemma 4.1 Let n ∈ N_{∞}. Then the sequence

0 → T → L(n)/(1 − σ)L(n) →^{\overline{μ}} D^{(n)} → 1,  (21)

where T is the torsion group of L(n)/(1 − σ)L(n), is exact. The homomorphism \overline{μ} is defined by the maps μ_{d} : G_{d} → D^{(n)}, a ↦ 1 − ϵ_{d}^{a} for d|n where ϵ_{d} is a primitive dth root of unity such that ϵ_{d} = ϵ_{t}^{t/d} whenever d|t.

Proof: Let N = ⨁_{d|n} M_{d} and Q = ∑_{d|n}⟨r + n_{d}(r); r ∈ E_{d}⟩. We write L(n) = N/Q. We will first show that \overline{μ} is well defined. Because of the isomorphism L(n)/(1 − σ)L(n) ≅ N/((1 − σ)N + Q) we have to show that (1 − σ)N and Q are mapped by μ to unit roots.

• For (1 − σ)N this follows from (1 − ϵ_{d}^{a})/(1 − ϵ_{d}^{−a}) = −ϵ_{d}^{a}.

• The generators of Q correspond to relations which arise by taking norms in cyclotomic fields. If N_{d,p} for p|d denotes the relative norm from Q(ϵ_{d}) to Q(ϵ_{d/p}) we have for d ≠ p that

N_{d,p}(1 − ϵ_{d}^{a}) = ∏_{{c∈G_{d} \atop c≡a mod d/p}}(1 − ϵ_{d}^{c}) = \{ \begin{matrix}1 − ϵ_{d/p}^{a} & if p^{2}|d, \\ (1 − ϵ_{d/p}^{a})/(1 − ϵ_{d/p}^{a′}) & if p^{2}|̸d \end{matrix}

(22)

with pa′ ≡ a mod d/p in the case p^{2}|̸d. Note that these relations can be derived directly from the polynomial identity ∏_{ν=0}^{p−1}(1 − xϵ_{p}^{ν}) = 1 − x^{p}. The module Q is generated by elements s(d, p, a) + n_{d}(s(d, p, a)). Noting that s(d, p, a) is mapped under μ on the product in the middle of (22) and −n_{d}(s(d, p, a)) is mapped to the right side of (22) the claim follows.

Up to now we have shown that \overline{μ} is well defined and obviously \overline{μ} is surjective. It remains to show that the kernel of \overline{μ} is indeed the torsion group of L(n)/(1 − σ)L(n). We look only at the case 2 < n < ∞. The case n = ∞ follows directly by reducing to the finite case. The cases n = 1 and n = 2 follow by direct computation. Because D^{(n)} is free it is sufficient to show that the rank of the free part of L(n)/(1−σ)L(n) is equal to the rank of D^{(n)}. In Lemma 5.2 we will show that rank(D^{(n)}) = \frac{1}{2}φ(n) + r − 1 where r denotes the number of primes dividing n. The module M_{+} is the free part of M/(1 − σ)M for any free module M, especially for M = L(n). We complete the proof by computing the rank of L(n)_{+}. First we observe that for any module M we have the formula rank(M_{+}) = \frac{1}{2}(rank(M) + m^{+}(M) − m^{−}(M)) and we apply it for M = L(n). Lemma 3.5 gives us the values m^{+}(L(n)) and m^{−}(L(n)). Because of Remark 2.5 the rank of L(n) is equal the sum of all ranks of the Y_{d} with d|n. These are listed in Table 2 (note that φ(n) = ∑_{d|n} g(d)). All this together gives rank(L(n)_{+}) = \frac{1}{2}φ(n) + r − 1. □

Lemma 4.2 L(n)_{+} ≅ D^{(n)}.

Proof: The claim is a direct consequence of Lemma 4.1. □ Using Lemma 4.2 we are now able to determine a basis of D^{(n)}. By means of weak σ-bases of the Y_{d} for d|n and (I) of section 2 bases of (Y_{d})_{+} can be constructed. With the help of Theorem 2.4 we construct a basis of L(n)_{+} which leads via the isomorphism of Lemma 4.2 to a basis of D^{(n)}. In the sequel we will give another approach using relative cyclotomic numbers that opens the way for the construction of bases for cyclotomic units. For n ∈ N let K^{(n)} = ∏_{d|n,d≠n} D^{(d)}. We call D̂^{(n)} = D^{(n)}/K^{(n)} the group of nth relative cyclotomic numbers.

Lemma 4.3 D̂^{(n)} ≅ (Y_{n})_{+} for n ≠ 4.

Proof: Setting N′ = ⨁_{d|n,d≠n} M_{d} and Q′ = ∑_{d|n,d≠n}⟨r + n_{d}(r); r ∈ E_{d}⟩ we obtain the commutative diagram

\begin{aligned}0 & → & (N′/Q′)_{+} & → & L(n)_{+} & → & (Y_{n})_{+} & → & 0 \\ & & ↓ & & ↓ & & ↓ \\ 0 & → & K^{(n)} & → & D^{(n)} & → & D̂^{(n)} & → & 0. \end{aligned}  (23)

The lower sequence is exact by definition. The upper sequence is exact because the same sequence without “+” is one of the sequences given in (9) which are defined by the MEn-system Γ(n) and these sequences have been shown to split over σ (this is not the case for n = 4, so we have to exclude this case in the theorem). All vertical arrows are surjective and the arrow in the middle is according to Lemma 4.2 an isomorphism. The left vertical arrow in the diagram is a restriction of this isomorphism and therefore also an isomorphism. With that also the right vertical arrow is an isomorphism.

□

Lemma 4.4 D̂^{(n)} is free for n ≠ 4.

Proof: Because (Y_{n})_{+} is free this follows directly from Lemma 4.3. □

Remark 4.5 Elementary calculations show that D̂^{(4)} is a group with two elements and therefore not free.

Theorem 4.6 Let n ∈ N_{∞}. If for each d|n, d ≠ 4 the set B̂_{d} ⊆ D^{(n)} induces a basis of D̂^{(d)} we have:

(a) ⋃_{d|n} B̂_{d} is a basis of D^{(n)} for n ≢ 0 mod 4.

(b) \{1 − ϵ_{4}\} ∪ ⋃_{{d|n \atop d≠2,4}} B̂_{d} is a basis of D^{(n)} for n ≡ 0 mod 4.

Proof: With Lemma 3.6 and Theorem 2.4 a basis of L(n)_{+} is given by the union of bases of the (Y_{d})_{+} where d|n with some slight modifications for n ≡ 0 mod 4. This leads to the theorem via the isomorphisms from Lemma 4.2 and Lemma 4.3. □

5 Cyclotomic Units

We write uniquely

n = q_{1} ⋯ q_{r} = p_{1}^{α_{1}} ⋯ p_{r}^{α_{r}}  (24)

with q_{i} = p_{i}^{α_{i}} for i = 1, …, r and p_{i} prime. Then the group of cyclotomic units C^{(n)} is modulo unit roots generated by

\frac{1 − ϵ_{q_{i}}^{a}}{1 − ϵ_{q_{i}}} where a ∈ G_{q_{i}} for i = 1, …, r,

and

1 − ϵ_{d}^{a} where 1 < d|n and d is not a prime power, a ∈ G_{d}.

The key for the construction of bases of cyclotomic units from bases of cyclotomic numbers is given in the next lemma.

Lemma 5.1 Let n ∈ N and B ⊆ C^{(n)}. Then B ∪ ⋃_{i=1}^{r}{1 − ϵ_{q_{i}}} is a basis of D^{(n)} if and only if B is a basis of C^{(n)}.

Proof: We show first the “only if”-case. Of course B is linearly independent in C^{(n)}. So it remains to show that B is a set of generators of C^{(n)}. We write u ∈ C^{(n)} as u = v ∏_{i=1}^{r}(1 − ϵ_{q_{i}})^{b_{i}} with v ∈ ⟨B⟩ and b_{i} ∈ Z. Taking the absolute norm of Q(ϵ_{n}) on both sides gives 1 = ∏_{i=1}^{r} p_{i}^{b_{i}k_{i}} with k_{i} = φ(n)/φ(q_{i}) ≠ 0 and it follows that b_{i} = 0 for all i ∈ {1, …, r}.

For the “if”-case we use the relation

∏_{{b∈Gq \atop b≡1 mod q/p}}(1 − ϵ_{q}^{b}) = 1 − ϵ_{q/p}  (25)

which holds for q = p^{α} with α > 1. Because of (1−ϵ_{q}^{a})/(1−ϵ_{q}) ∈ C^{(n)} the set B ∪ ⋃_{i=1}^{r}{1 − ϵ_{q_{i}}} is a set of generators of D^{(n)}. The linearly independence of the 1 − ϵ_{q_{i}} and B follows by taking norms as in the “only if”-case. □

Lemma 5.2 Let n > 2. The rank of D^{(n)} is \frac{1}{2}φ(n) + r − 1.

Proof: The group of cyclotomic units has finite index in the full unit group of Q(ϵ_{n}) (see [7]). Therefore we have rank(C^{(n)}) = \frac{1}{2}φ(n) − 1 ([9], Proposition 7-6-1). The claim follows from Lemma 5.1. □ We define the relative cyclotomic units by Ĉ^{(n)} = C^{(n)}/L^{(n)} with L^{(n)} = ∏_{d|n, d≠n} C^{(d)}. The connection between relative cyclotomic units and relative cyclotomic numbers is given by the isomorphisms

Ĉ^{(n)} ≅ D̂^{(n)} if n is not a prime power,

Ĉ^{(q)} ≅ ⟨\frac{1 − ϵ_{q}^{a}}{1 − ϵ_{q}} K^{(q)}; a ∈ G_{q}⟩ ≤ D̂^{(q)} if n = q is a prime power.

Analogously to Theorem 4.6 we obtain:

Theorem 5.3 Let n ∈ N_{∞}. If B̂_{d} ⊆ C^{(n)} induces a basis of Ĉ^{(d)} for d|n then B_{n} = ⋃_{d|n} B̂_{d} is a basis of C^{(n)}.

Proof: It is enough to consider the case when n is finite. The case n = ∞ follows directly from the finite case. Let p be a prime and α > 0. Because of C^{(p^{α−1})} = L^{(p^{α})} the sequence

1 → C^{(p^{α−1})} → C^{(p^{α})} → Ĉ^{(p^{α})} → 1  (26)

is exact. The group Ĉ^{(p^{α})} is free, because it is isomorphic to a subgroup of D̂^{(p^{α})} which is free for p^{α} ≠ 4 and directly from the definition follows Ĉ^{(4)} = 1. By induction we deduce the claim of the theorem for n = p^{α}. From Lemma 5.1 it follows that E_{q} = {1 − ϵ_{q}} ∪ B_{q} is a basis of D^{(q)}.

Let Θ_{n} be the set of all divisors of n which are not prime powers. By Theorem 4.6 there exist F̂_{d} ⊆ D^{(n)} with the following properties:

(a) F_{n} = ⋃_{d|n} F̂_{d} is a basis of D^{(n)},

(b) F_{q_{i}} = ⋃_{d|q_{i}} F̂_{d} is is a basis of D^{(q_{i})} for i = 1, …, r,

(c) F̂_{d} = B̂_{d} for d ∈ Θ_{n}.

By (c) we get F_{n} = ⋃_{i=1}^{r} F_{q_{i}} ∪ ⋃_{d∈Θ_{n}} B̂_{d} which is by (a) a basis of D^{(n)}. Using (b) we can exchange F_{q_{i}} by E_{q_{i}} and obtain

⋃_{i=1}^{r} E_{q_{i}} ∪ ⋃_{d∈Θ_{n}} B̂_{d} = ⋃_{i=1}^{r}{1 − ϵ_{q_{i}}} ∪ ⋃_{d|n} B̂_{d} = ⋃_{i=1}^{r}{1 − ϵ_{q_{i}}} ∪ B_{n}  (27)

as basis of D^{(n)}. With Lemma 5.1 the claim follows. □ For n ∈ Θ_{n} we have

Z(n)_{+} = (Y_{n})_{+} ≅ D̂^{(n)} ≅ Ĉ^{(n)}.  (28)

A basis of Z(n)_{+} follows from a weak σ-basis of Z(n). A weak σ-basis of Z(n) can be constructed with the help of Table 1 and method (III) of section 2. In order to complete the construction of a basis of C^{(n)} it remains to give a basis for Ĉ^{(q)} when q = p^{α}. For α = 1 it is well known [8] that

\{\frac{1 − ϵ_{p}^{a}}{1 − ϵ_{p}}; 1 < a < p/2\}  (29)

induces a basis of Ĉ^{(p)}. For α > 1 let Γ = \{b ∈ G_{q/p}; 1 ≤ b ≤ \frac{1}{2}q/p\}. Then

\{\frac{1 − ϵ_{q}^{ap^{α−1}+b}}{1 − ϵ_{q}}; (b, a) ∈ Γ × \{1, …, p − 1\}\}  (30)

induces a basis of Ĉ^{(q)}.

6 Remarks

The aim of this paper is the construction of bases for cyclotomic units. However there are further applications concerning the cyclotomic module and weak σ-bases. We give here two examples. For details see [1].

• A similar construction as for the group of cyclotomic units can be done for the Stickelberger ideal. Let I_{n} the ideal generated by the Stickelberger elements θ(a) = ∑_{τ∈G_{n}}⟨−aτ/n⟩τ^{−1} and ω_{n} = ΣG_{n} for n odd and ω_{n} = \frac{1}{2}ΣG_{n} for n even. Then we have in analogy to Lemma 4.2 the isomorphism L(n)_{−} ≅ I/⟨ω_{n}⟩ via G_{d} ∋ a ↦ θ(an/d).

• A stronger definition of weak σ-bases can be used to construct explicitly relations in the group of cyclotomic units, especially the relations not arising from norm relations or complex conjugation which has been discovered first by Ennola [2]. These relations are in our context implicitly given by the set T in the exact sequence (21).

References

[1] M. Conrad, “Basen von Moduln mit Anwendung auf Kreiseinheiten und Stickelbergerelemente,” Dissertation an der Universität des Saarlandes, Saarbrücken, 1997.

[2] V. Ennola, On Relations between Cyclotomic Units, J. Number Theory 4 (1972), 236-247.

[3] R. Gold and J. Kim, Bases for cyclotomic units, Compositio Math. 71 (1989), 13-28.

[4] R. Kučera, On bases of the Stickelberger ideal and of the group of circular units of a cyclotomic field, J. Number Theory 40 (1992), 284-316.

[5] K. Ramachandra, On the units of cyclotomic fields, Acta Arith. 12 (1966), 165–173.

[6] C.-G. Schmidt, Die Relationsfaktorgruppen von Stickelberger-Elementen und Kreiszahlen J. Reine Angew. Math. 315 (1980), 60–72.

[7] W. Sinnott, On the Stickelberger ideal and the circular units of a cyclotomic field, Ann. Math. 108 (1978), 107-134.

[8] L. C. Washington, “Introduction to Cyclotomic Fields,” Springer, New York 1982.

[9] E. Weiss, “Algebraic number theory,” McGraw-Hill, New York 1963.