Constructible Units in Abelian p-Group Rings

by

Klaus Hoechsmann and Jürgen Ritter

Abstract: Let A be a finite abelian group of exponent p^{m} > 1, an odd prime power, and consider the group Ω(A) of units in the group ring ZZA which appear circular in each cyclotomic Wedderburn component. We explicitly construct generators for a subgroup Λ(A) ⊆ Ω(A) of p-power index, which equals Ω(A) if and only if p is regular. Moreover, we determine the structure of Λ(A) as a G-module, where G is the unit group of ZZ mod p^{m}ZZ.

0. Introduction.

The aim of this paper is to exhibit and investigate a certain “large” group of explicitly identifiable units in the group ring ZZA of a finite abelian p-group A, where p is an odd prime.

To begin with consider the polynomials f_{c}(x) = 1 + x + ⋯ + x^{c−1}, defined for positive integers c, and note the elementary cocycle relation

f_{c}(x)f_{b}(x^{c}) = f_{cb}(x).

In the sequel x will always be specialized to an element of finite multiplicative order, say x^{q} = 1, in some ring R, and b, c will be a pair of reciprocals modulo q, say cb = 1 + kq. Then xf_{q}(x) = f_{q}(x), and the cocycle relation turns into

f_{c}(x)f_{b}(x^{c}) = 1 + kf_{q}(x).

Now, if x − 1 is not a zero-divisor in R, it follows that f_{q}(x) vanishes and therefore f_{c}(x) is a unit in R. If f_{q}(x) ≠ 0 this fails; however, a simple computation shows that

e_{c,b}(x) = f_{c}(x)f_{b}(x) − kf_{q}(x)  (†)

is always a unit; its inverse is f_{c}(x^{b})f_{b}(x^{c}) − kf_{q}(x). Although e_{c,b} depends only on the inverse pair c, b mod q, we keep the somewhat redundant double subscript as a reminder of its make-up.

For use with the abelian p-group A, we shall fix a q = p^{m} such that A^{q} = 1, and modify the elements f_{c}(x) and e_{c,b}(x) by suitable powers of x so as to render them symmetric, i.e. invariant under the involution ⋆ : ZZA → ZZA defined by x^{⋆} = x^{−1} for x ∈ A. Explicitly, we replace f_{c}(x) by v_{c}(x) = x^{ϵ(c)}f_{c}(x), the exponents ϵ(c) = \frac{1−c}{2} to be read modulo q, and use

w_{c,b}(x) = v_{c}(x)v_{b}(x) − kf_{q}(x)  (‡)

instead of e_{c,b}(x). The motive for this symmetrization is convenience rather than necessity. It is well-known (cf. [7], for instance) that the unit group U(A) of ZZA is a direct product U(A) = ±A × S(A), whose first factor is exactly the torsion subgroup, and whose second factor S(A) consists of the symmetric units with coefficient-sum (“augmentation”) = 1. In studying U(A) it is therefore reasonable to leave aside the torsion and concentrate on the subgroup S(A).

Now fix a positive g ∈ ZZ so that its image in G = (ZZ/qZZ)^{×} generates G, and choose positive h, k ∈ ZZ such that gh = 1 + kq. These choices determine a polynomial w(x) = w_{g,h}(x) according to (‡). The subgroup Λ(A) ⊂ S(A) generated by elements of the form w(x) with x ∈ A is called the group of constructible units in ZZA. It is very transparent. First, it turns out to be the direct product of groups W (C) corresponding to the cyclic subgroups C of A. Second, each W (C) is generated by the elements w(x) for x ∈ C of maximal order, and the only relation (apart from w(x) = w(x^{−1})) between these generators is that their product is 1. This is essentially the contents of Theorems 2.3 and 3.5 below.

A unit in the ring ZZ[exp 2πi/q] of cyclotomic integers will be called circular if it is a product of elements of the form v_{c}(ξ), with ξ a q-th root of 1. Such units are most often referred to as “cyclotomic”, but this term is already burdened with more general meanings. The label “circular” will also be applied to any symmetric unit of ZZA of augmentation 1 which is mapped to circular units by all characters ψ : A → C^{×}. The group Ω(A) ⊆ S(A) of circular units contains Λ(A) as a large subgroup in the sense that ψ(Λ(A)) = ψ(Ω(A)) for all characters ψ. It will be shown in section 3 that the index [Ω(A) : Λ(A)] is a p-power, and Ω(A) = Λ(A) if and only if p is regular. In turn, the index [S(A) : Ω(A)] is prime to p, provided only that p be semi-regular (a mild condition to which there are no known counterexamples).

To prove these results, we first have to establish them for the special case of cyclic A. This will be done in section 2. In section 1 we must deal with another preliminary question: which circular units in the cyclotomic ring ZZ[exp 2πi/q] can be lifted to units in ZZA ?

1. Lifting Circular Units.

Retaining the notation introduced above, fix a cyclic group C of order p^{m} > 1 and a generator x of C. With ζ denoting a primitive p^{m}-th root of 1, consider the prime element P = ζ^{1/2} −ζ^{−1/2} in the ring ZZ[ζ] of cyclotomic integers (exponents to be read in ZZ/p^{m}ZZ). Since P is transformed to −P by complex conjugation, any unit of the form u(σ) = P^{σ−1}, with σ in the Galois group G, is real. The group Ω(ζ) generated by these is known as the group of (real) circular units; in other words,

Ω(ζ) = P^{ΔG},

where ΔG denotes the kernel of the augmentation map ZZ(G) ⟶ ZZ. It is of finite index in the full unit group U(ζ) of ZZ[ζ]; (cf. [10], Theorem 8.2).

The question to be treated in this section is: which elements of Ω(ζ) appear in the image of S(C) under the ring surjection φ : ZZ(C) ⟶ ZZ[ζ] induced by φ : x ↦ ζ? We shall find that they are precisely the elements of Ω(ζ)^{ΔG}. In preparation of the proof, note that the obvious cocycle relation

u(στ) = u(σ)^{τ} u(τ) = u(σ)u(τ)u(σ)^{τ−1}  (1)

yields a group epimorphism

u : G ⟶ Ω(ζ)/Ω(ζ)^{ΔG},  (2)

which shows that the factor group is cyclic of order dividing p^{m−1}(p − 1). Eventually it will be seen that (2) is an isomorphism.

Of course, G can be canonically identified with the unit group of ZZ/p^{m}ZZ. Indeed, any positive integer c prime to p yields a σ(c) ∈ G = Aut(C) = Aut(ZZ[ζ]) by σ(c) : x ↦ x^{c}, for x ∈ C and σ(c) : ζ ↦ ζ^{c}. Instead of u(σ(c)) we write u(c), so that

u(c) = v_{c}(ζ).

Proposition 1.1: Ω(ζ)^{ΔG} ⊆ φ(S(C)). Proof. For any c prime to p and τ ∈ G, we wish to construct a φ-preimage of u(c)^{τ−1} in S(C). Choose b such that bc = 1 + kp^{m} for some positive integer k. The cocycle relation (1) shows that 1 = u(bc) = u(b)^{σ(c)}u(c), so that u(c)^{−1} = u(b)^{σ(c)}. The element w_{c,b,τ} (x) ∈ ZZ(C) given by

w_{c,b,τ} (x) = v_{c}(x^{τ})v_{b}(x^{c}) − kf_{q}(x)  (3)

has augmentation 1 and maps to u(c)^{τ−1} under φ. Indeed the second term of this expression vanishes under any ψ : ZZ(C) ⟶ ZZ[ζ] with ψ(C) ≠ {1}, showing that ψ(w_{c,b,τ} (x)) is a circular unit in every Wedderburn component of ZZ(C). Q.E.D

We note in passing that the units given in (‡) relate to these new ones as follows:

w_{c,b}(x) = w_{c,b,σ(c)}(x^{b}); thus w_{c,b}(ζ) = u(c)^{1−σ(b)} = u(c)u(b).  (4)

Definition 1.2: A unit of u ∈ ZZ[ζ] is called liftable, if there is a w ∈ S(C) such that u = φ(w); it is called p-liftable, if there is a w_{p} ∈ S′_{p}(C) such that u = φ(w_{p}), where S′_{p}(C) is the group of symmetric units in the p-adic group ring ZZ_{p}(C) with augmentation 1 and G-norm 1.

The subgroups of liftable and p-liftable elements of Ω(ζ) will be denoted by L(ζ) and L_{p}(ζ), respectively.

In view of Proposition 1.1, we have

Ω(ζ)^{ΔG} ⊆ L(ζ) ⊆ L_{p}(ζ),

the second inclusion following from the obvious fact that S(C) is contained in the group S′_{p}(C). To show that these are really equalities, we consider the composite group epimorphism

G⟶^{u}Ω(ζ)/Ω(ζ)^{ΔG} ⟶ Ω(ζ)/L_{p}(ζ),  (5)

and shall prove its injectivity.

Theorem 1.3: L_{p}(ζ) = L(ζ) = Ω(ζ)^{ΔG} Proof. As promised, we shall prove the injectivity of the map (5), to wit: if u(c) is pliftable, then c ≡ 1 mod p^{m}. We must, however, make better use of the fact that elements of S′_{p}(C) have augmentation 1, in particular, that L_{p}(ζ) = Ω(ζ) ∩ φ(S′_{p}(C)) is contained in J(ζ), the subgroup of Ω(ζ) consisting of elements ≡ 1 modulo (ζ − 1). The exact sequence

1 ⟶ J(ζ) ⟶ Ω(ζ) ⟶ IF_{p}^{×} ⟶ 1

shows that [Ω(ζ) : J(ζ)] = p − 1. Since u(c) ≡ c mod(ζ − 1) and L_{p}(ζ) ⊂ J(ζ), there is no hope for u(c) to be p-liftable, unless c ≡ 1 modp. For the remainder of the proof, we may therefore assume that m ≥ 2.

Now, u(c) ∈ L_{p}(ζ) implies u(c^{n}) ∈ L_{p}(ζ), for any n, because u(c^{n}) ≡ u(c)^{n} modulo Ω(ζ)^{ΔG}. Since c = 1 + ap, we can obviously find an n such that c^{n} = 1 mod p^{m−1}. Thus, if c ≢ 1 mod p^{m}, we may assume that c = 1 + ep^{m−1}, with 0 < e < p.

The kernel of φ : ZZ_{p}(C) ⟶ ZZ_{p}[ζ] is the ideal ZZ_{p}(C) · (1 + x^{n} + … + x^{n(p−1)}), where n = p^{m−1} and x generates C. Hence, p-liftability of u(c) entails the existence of k(x) ∈ ZZ_{p}(C) such that

ṽ = v_{c}(x) − k(x)(1 + x^{n} + … + x^{n(p−1)}) ∈ S′_{p}(C).

For μ = 0, 1, …, m, let ψ_{μ} : ZZ_{p}(C) ⟶ ZZ_{p}[ζ_{μ}] be defined by ψ_{μ}(x) = ζ_{μ} = ζ^{p^{m−μ}}. So, ψ_{m} = φ and ψ_{0} is the augmentation map. The Wedderburn decomposition

Q_{p}(C)⟶^{∼}∑_{μ=0}^{m} Q_{p}(ζ_{μ})

shows that the condition N_{G}(ṽ) = 1 is equivalent to the m + 1 conditions N_{G}(ψ_{μ}(ṽ)) = 1. Now, φ(ṽ) = u(c) = P^{σ(c)−1} obviously satisfies this for μ = m. For μ = 0 we have φ_{0}(ṽ) = c − pk(1) = 1 automatically because ṽ ∈ S_{p}(C); hence k(1) = ep^{m−2}. Thus the brunt of the norm condition falls on the components ψ_{μ}(ṽ) for μ = 1, …, m − 1, where v_{c}(ζ_{μ}) = 1 because c ≡ 1 modulo p^{μ}, and therefore ψ_{μ}(ṽ) = ψ_{μ}(1 − pk(x)). Since, for μ > 0, the norm N_{G} is the p^{m−μ}-th power of the field-norm N_{μ} in Q_{p}(ζ_{μ})/ Q_{p}, we have to show that the conditions N_{μ}(ψ_{μ}(1 − pk(x))) = 1 for μ = 1, …, m − 1 are incompatible with k(1) = ep^{m−2}.

For brevity let us write 1 − pk(x) = l(x). The Q_{p}-representation l(x) ⟼ [l(1), l(ζ_{1}), …, l(ζ_{m−1})] is evidently isomorphic to the regular representation of l(x^{p}) ∈ Q_{p}(C^{p}), and

the product ∏_{μ=0}^{m−1}N_{μ}(l(ζ_{μ})) is equal to the determinant of the latter. Our norm condition

thus forces this determinant to equal l(1) = 1−ep^{m−1}. Putting y = x^{p} and l(y) = a_{0}+a_{1}y+ …+a_{n−1}y^{n−1}, we see that the columns of the corresponding matrix are cyclic permutations of the n-tuple [a_{0}, a_{1}, …, a_{n−1}]. Since l(y) = 1 − pk(y), we have a_{i} ≡ 0 mod p for i > 0, and the determinant is ≡ a_{0}^{n} modulo p^{m} by Lemma 1.4 below. But a_{0} ≡ 1 mod p implies a_{0}^{n} ≡ 1 mod p^{m}, in contradiction to the value 1 − ep^{m−1} imposed by the norm condition.

The proof of Theorem 1.3 is now complete except for a lemma on the kind of matrices, often called “circulants”, encountered in regular representations of cyclic group-rings. We shall present it in somewhat greater generality than required above.

Let R_{n} = ZZ/nZZ for some integer n > 1. Given an n-tuple [a_{0}, a_{1}, …, a_{n−1}] of elements from some commutative ring, define the circulant C(a_{0}, …, a_{n−1}) to be the n × n matrix (a_{i−j}), indices to be read in R_{n}.

Lemma 1.4: Let p be a prime, and suppose that ord_{p}n = m − 1, m ≥ 1. Then, for indeterminates t_{0}, …, t_{n−1}, we have

detC(t_{0}, pt_{1}, …, pt_{n−1}) ≡ t_{0}^{n} mod p^{m}.

Proof. Note that the additive group R_{n} acts, by conjugation, on the symmetric group S_{n}. Explicitly, for k ∈ R_{n} and σ ∈ S_{n}, σ^{k}(ν + k) = σ(ν) + k. The generic term (−1)^{sgnσ}t_{σ} = (−1)^{sgnσ} ∏_{ν=0}^{n−1}t_{ν−σ(ν)} of detC(t_{0}, t_{1}, …, t_{n−1}) is obviously invariant under this action, so that we can write

detC(t_{0}, …, t_{n−1}) = ∑_{H}(−1)^{sgnH}|H|t_{H}

where H runs over the R_{n}-orbits of S_{n}.

Let us consider a fixed H, generated by some σ ∈ S_{n}. Suppose t_{H} = t_{0}^{n−d}t_{i_{1}}^{r_{1}} … t_{i_{s}}^{r_{s}}, with r_{1} +…+r_{s} = d ≥ 1 and i_{1}, …, i_{s} non-zero. For the subsets F_{i} = {ν ∈ R_{n}|ν −σ(ν) = i} and S(σ) = {k ∈ R_{n}|σ^{k} = σ} of R_{n}, we obviously have F_{i} + S(σ) ⊆ F_{i}, so that r_{1} = |F_{i_{1}}| ≥ |S(σ)|. Putting r = |S(σ)|, the inequality d ≥ r_{1} ≥ r now implies d > ord_{p}r. Indeed, c = ord_{p}r ≥ d would entail r = lp^{c} ≥ p^{d} > d.

Since |H| = n/r, the corresponding term of detC(t_{0}, pt_{1}, …, pt_{n−1}) is ±|H|p^{d}t_{H} which turns out to be ≡ 0 mod p^{m}, because ord_{p}|H|p^{d} = ord_{p}n + d − ord_{p}r ≥ ord_{p}n + 1 = m. Q.E.D.

Remark 1.5: The reader may have wondered why we use G instead of the more natural group G_{+} = G/ < ⋆ >, which is the Galois group of the real subfield of Q(ζ). The reason is that P is not in this field. Had we used the real prime P^{2} = (ζ − 1) + (ζ^{−1} + 1), we would have wound up with the smaller group P^{2ΔG_{+}} = Ω(ζ)^{2}. As far as the norm condition is concerned, it comes out equally using G or G_{+}, because the G-norm is the square of the G_{+}-norm and there is no 2-torsion in S(C). We shall continue to use both groups side-by-side (G_{+} rarely), making no notational distinction between elements of G (except ⋆ ) and their images in G_{+}.

2. Generating Circular Units.

In this section, we continue with the cyclic group C of order q = p^{m}, investigating the structure of the group Λ(C) of constructible units and its relation to the group Ω(C) of circular ones. Fixing a generator x of C, we let w = w(x) = v_{g}(x)v_{h}(x) − kf_{q}(x) be the unit defined in the introduction. G = Aut(C) is generated by σ = σ(g) as in the last section.

It is time to have a closer look at the Wedderburn components of the special unit w, i.e. its images under ψ_{μ} : S(C) ⟶ S(ζ_{μ}) with ψ_{μ}(x) = ζ_{μ} = ζ^{p^{m−μ}}, for μ = 0, …, m. Of course, ψ_{0}(w) = 1 and ψ_{m}(w) = φ(w). By (4), we have φ(w^{σ}) = u(σ)^{σ−1}. The following lemma shows how the various components ψ_{μ}(w) are related to one another.

Lemma 2.1: For 2 ≤ μ ≤ m,

ψ_{μ−1}(w) = ψ_{μ}(w)^{s_{m−μ+1}},  (6)

where s_{ν} = 1 + τ_{ν} + … + τ_{ν}^{p−1} with τ_{ν} ∈ G of order p^{ν}. Proof. By an induction we need not formalize it suffices to show that, for m ≥ 2, we have ψ_{m−1}(w) = ψ_{m}(w)^{s_{1}}, s_{1} = 1+τ_{1}+…+τ_{1}^{p−1} ∈ ZZ(G) with τ_{1} ∈ G of order p. In other words, ψ_{m−1}(w) is the Galois norm of w in the extension Q(ζ)/ Q(ζ_{m−1}). Since m > 1, we have ψ_{m−1}(w) = ψ_{m−1}(v_{g}(x))ψ_{m−1}(v_{h}(x)); the term kf_{q}(x) vanishes under ψ_{m−1}. It suffices therefore to prove ψ_{m−1}(v) = ψ_{m}(v)^{s_{1}} for the non-units v = v_{c}(x) = x^{\frac{1−c}{2}} (1+x+…+x^{c−1}).

A glance at the minimal polynomial of ζ over Q(ζ_{m−1}) shows that ψ_{m−1}(x) = ζ_{m−1} = ζ^{s_{1}} = ψ_{m}(x)^{s_{1}}. Substituting 1 in the factorization of the same polynomial, we get

ψ_{m−1}(1 − x) = 1 − ζ_{m−1} = (1 − ζ)^{s_{1}},

and finally

ψ_{m−1}(1 + x + … + x^{c−1}) = \frac{1 − ζ_{m−1}^{c}}{1 − ζ_{m−1}} = (\frac{1 − ζ^{c}}{1 − ζ} )^{s_{1}} = ψ_{m}(1 + … + x^{c−1})^{s_{1}},

finishing the proof.

Definition 2.2: An element u ∈ S(C) is called uniform if φ(u) ∈ Ω(ζ) and every ψ_{μ}(u) is the Q(ζ)/ Q(ζ_{μ})-norm of φ(u). The group of uniform units will be denoted by W (C).

Theorem 2.3: W (C) is generated over ZZG by w, and is mapped isomorphically onto L(ζ) by φ. Mapping w to 1, we get an isomorphism

W (C) ⟶ ZZG_{+}/(t_{+}),

where t_{+} is the sum over all elements of G_{+}. Proof. By (6) and the “telescoping” of norms, w is uniform and w^{ZZG} ⊆ W (C). Since σ = σ(g) generates G, Theorem 1.3 shows that the unit φ(w^{σ}) = u(σ)^{σ−1} generates L(ζ) over ZZ(G). Hence w^{ZZG} is mapped onto L(ζ).

As every element of W (C) is determined by its φ-image, it follows that the restriction of φ to W (C) is injective and that w^{ZZG} = W (C).

For the second statement we must show that L(ζ) ≃ ZZ(G_{+})/(t_{+}). To see this, remember that L(ζ) is generated by u(σ)^{σ−1}. If u(σ)^{(σ−1)α} = 1 with α ∈ ZZG_{+}, we must show that α is a multiple of t_{+}. Now u(σ)^{α(σ−1)} = 1 implies u(σ)^{α} ∈ Q, hence u(σ)^{α} = ±1. Reducing mod t_{+} we may assume that α = ∑_{τ≠1}a_{τ} τ, i.e. that the |G_{+}| − 1

conjugates u(σ)^{τ} , τ ≠ 1, are dependent modulo {±1}. This is impossible since they must span a lattice of rank |G_{+}| − 1. Q.E.D.

Remark 2.4: If γ = |G| and (c, p) = 1, we have c^{γ} = 1 + k_{c}q for suitable k_{c} ∈ ZZ, and the element w′_{c}(x) = v_{c}(x)^{γ} − k_{c}f_{q}(x) is obviously in Ω(C). These are essentially the units used by Bass in [1]. The proof of Lemma 2.1 shows that they are uniform. Hence the group W′(C) generated by them is isomorphic to its φ-image, which is generated over ZZ(G) by u(σ)^{γ}. In particular, [W (C) : W′(C)] = γ^{r−1}, where r is the rank of W (C).

To get more mileage out of W (C), we enlarge our ring ZZ(G) of operators, adjoining to it the Frobenius map π : C ⟶ C^{p} with π(x) = x^{p}. What we get is the monoid-ring ZZ(M) = ZZ(G, π), where M is the multiplicative monoid End(C), which is generated by G and π. In more familiar terms, ZZ(M) is a kind of polynomial ring in π with coefficients in ZZ(G), and with relations π^{m} = 0 and π^{ν}(τ_{ν} − 1) = 0, where τ_{ν} ∈ G has order p^{ν}(ν = 1, …, m − 1). ZZ(M) does not operate on U(C) because π^{m} induces the (non-trivial) augmentation map, but on S(C) everything works. To forestall any confusion between 0 and 1, we write operators from ZZ(M) as exponents, so u^{π^{m}} = u^{0} = 1 looks natural.

Since φ(a^{π}) = ψ_{m−1}(a) for any a ∈ ZZ(C), it is clear, in view of (3) and Proposition 2.3, that W (C)^{π} = W (C^{p}) and so on. For w ∈ W (C) formula (6) translates into φ(w^{π^{ν}}) = φ(w^{s_{ν}π^{ν−1}}), or

φ(W (C)^{π^{ν}−s_{ν}π^{ν−1}}) = 1  (7)

for ν = 1, …, m − 1. This will be crucial in the following argument.

Lemma 2.5: The natural map of the direct product

λ : ∏_{ν=0}^{m−1} W (C^{p^{ν}}) ⟶ Ω(C)

is an injection with finite cokernel. Proof. Since both sides are free ZZ-modules of the same rank \frac{1}{2}(p^{m} − 1) − m, it suffices to prove finiteness of the index. Note that the image of this map is w^{ZZ(M)}, where w is any ZZ(G)-generator of W (C). The proof will be by induction, with nothing to do if m = 1.

By Theorem 2.3, the surjection φ : Ω(C) ⟶ L(ζ) splits: Ω(C) = W (C) × kerφ. We shall be done if we can prove that the elements w^{π^{ν}−s_{ν}π^{ν−1}}, ν = 1, …, m − 1, generate over ZZ(G) a submodule of finite index in kerφ, cf. (7). For this purpose we introduce the map

κ : W × W^{π} × … × W^{π^{m−2}} ⟶ kerφ,

whose components are (π − s_{1}), …, (π − s_{m−1}).

At this point we need to refer to the fundamental pull-back diagram of ring maps

ZZ(C) ⟶^{φ} ZZ[ζ]
π ↓  ↓
ZZ(C^{p}) ⟶^{ρ} IF_{p}(C^{p}),

(8)

in which IF_{p} = ZZ/pZZ and ρ denotes reduction of coefficients (cf. [8], §1). It is easy to see that the pull-back property is preserved when we apply the functors U, S, Ω. In each case, it shows that π induces an isomorphism kerφ⟶^{∼} kerρ. We shall use it for Ω to produce a commutative square

\begin{matrix}W × W^{π} × … × W^{π^{m−2}} & ⟶^{κ} & kerφ \\ π ↓ & & ↓ π \\ Ω(C^{p}) & ⟶^{π−p} & kerρ \end{matrix}  (9)

whose right vertical arrow is an isomorphism, thus shifting the problem to showing the clockwise composite πκ is of finite index. Commutativity of this diagram is due to the fact that s_{1}π = pπ, etc. It allows us to consider the counter-clockwise composite π(π − p) instead.

Now, the left vertical arrow has finite index by induction hypothesis. The bottom arrow π − p has finite index for reasons of rank, because

Ω(C^{p})⟶^{π−p} kerρ ↪ Ω(C^{p})

is injective. Indeed u^{π} = u^{p} implies 1 = u^{π^{m}} = u^{p^{m}}, but there is no p-torsion in S(C^{p}). This finishes the proof.

Now that we know that im(λ) is a direct product, we can dispense with λ and consider Λ(C) = W (C)×W (C)^{π} ×…×W (C)^{π^{m−1}} a subgroup of Ω(C). Note that Λ(C^{p}) = Λ(C)^{π} because of the similar equation for W.

Lemma 2.6: The index [Ω(C) : Λ(C)] is a power of p. Proof. It will be seen by a simultaneous induction that each of the maps (i)π : Ω(C) ⟶ Ω(C^{p}), (ii) π − p : Ω(C) ⟶ kerρ, and (iii) λ itself, has p-power index.

Since W (C^{p}) = W (C)^{π}, the induction hypothesis on (iii) implies that [Ω(C^{p}) : W (C)^{πZZ(M)}] is a p- power, hence (i).

For (ii), let u ∈ Ω(C) with ρ(u) = 1. By induction assumption, there exists a v_{1} ∈ Ω(C^{p}) such that u^{πp^{k}} = v_{1}^{π−p} for suitable k. By (i), there exists a v ∈ Ω(C) such that v^{π} = v_{1}^{p^{l}} for suitable l. Hence u^{p^{k+l}π} = v_{1}^{p^{l}(π−p)} = v^{π(π−p)}, and (u^{p^{k+l}}v^{p−π})^{π} = 1. Therefore

(u^{p^{k+l}}v^{p−π})^{p−π} = u^{p^{k+l+1}}v^{p(p−π)},

i.e. a certain p-power of u is in Ω(C)^{π−p}, whence (ii).

For (iii), we go through the proof of Lemma 2.5 once more, replacing the words “finite index” by “p-power index”.

Remark 2.7: It is a classical fact (cf. [10], Theorem 8.2) that [S(ζ_{μ}) : Ω(ζ_{μ})] = h_{μ}^{+}, the class-number of Q(ζ_{μ} + ζ_{μ}^{−1}). Putting h^{+}(m) = h_{1}^{+} … h_{m}^{+}, it is clear that S(C)^{h^{+}(m)} ⊆ Ω(C). Hence [S(C) : Ω(C)] is finite and involves only prime divisors occuring in h^{+}(m). If h_{1}^{+} is not divisible by p, the prime p is called semi-regular. In that case h^{+}(m) and hence [S(C) : Ω(C)] is also prime to p. Vandiver’s Conjecture – cf. [10], Remark p. 159 – says that this would be true for any prime p. So far it has been verified for p < 125000.

Remember that p is regular if the class number of the full cyclotomic field Q(ζ_{1}) is prime to p. This is definitely a restrictive condition. The first irregular prime is 37.

Lemma 2.8: If p is a regular prime, Λ(C) = Ω(C). Proof. The main reason for this is a generalization of Kummer’s Lemma (cf. [5]), which for regular p says that u ↦ u^{π−p} yields an isomorphism

S(C)⟶^{∼} kerρ,

ρ now referring to the reduction map on all of S(C). Shifting our attention from Ω(C) to S(C) is not difficult, because, by Remark 2.7 and a well-known property of regular primes (cf. [10]), Corollary 10.5), the index [S(C) : Ω(C)] is prime to p.

For any ZZ-module X, let X̂ = X ⊗ ZZ_{p}. Then we have Ω̂(C) = Ŝ(C). Since ZZ_{p} is flat over ZZ, the hat functor is exact. Thus it preserves the isomorphisms Ŝ(C)⟶^{∼} kerρ̂, as well as kerφ̂⟶^{π} kerρ̂ (in fact it preserves pull-backs). We can now read through the proof of Lemma 2.5 for a third time, putting hats on everything and replacing the words “finite index” by “index 1”. This proves that λ̂ is an isomorphism, hence that the index of λ is prime to p, Q.E.D.

Lemma 2.9: If m ≥ 2 and Λ(C) = Ω(C), then p is regular. Proof. We have the direct decomposition Ω(C) = W (C)× kerφ and Λ(C) = W (C)×imκ, where κ is as defined in the proof of Lemma 2.5. Hence Ω(C)/Λ(C) is isomorphic to kerφ/imκ. A glance at diagram (9) shows that π induces a surjection of kerφ/imκ onto kerρ/Ω(C^{p})^{π−p}. Hence Ω(C) = Λ(C) implies kerρ = Ω(C^{p})^{π−p}.

Now we have both maps into kerρ in diagram (9) being isomorphisms. Hence the surjectivity of κ implies that of π : W × … × W^{π^{m−2}} ⟶ Ω(C^{p}), i.e. Ω(C^{p}) = Λ(C^{p}).

By recursively applying this fact, we may assume that m = 2. Then C^{p} has order p, and the result on kerρ yields an exact sequence

1 ⟶ Ω(C^{p})^{p} ⟶ Ω(C^{p})⟶^{ρ}\overline{S}(C^{p}),

i.e. the image ρ(Ω(C^{p})) has dimension \frac{p−3}{2} in the IF_{p}-module \overline{S}(C^{p}). It is well-known that this is precisely equivalent to the non-vanishing mod p of the Bernoulli numbers B_{2}, B_{4}, …, B_{p−3}, i.e. to the regularity of p (cf. [2], Ch. V, §6.3, and [8], §6) Q.E.D.

3. Units in Abelian p-Group Rings.

Finally we can deal with a general abelian p-group A of exponent q = p^{m}. We shall temporarily have to do with the maximal order M_{A} in the rational group algebra Q(A). A unit of M_{A} will be called circular if it is symmetric of augmentation 1, and appears as a circular unit in every Wedderburn component of Q(A). The groups of circular units in ZZ(A) and in the maximal order will be denoted by Ω(A) and Ω(M_{A}), respectively. Between them lies the group Ω^{1}(M_{A}) of those circular units which are ≡ 1 modulo the ideal Δ(A)M_{A}, where Δ(A) is the augmentation kernel in ZZ(A). Its Wedderburn components Ω^{1}(ζ_{μ}) ⊂ Ω(ζ_{μ}) consist of units which are ≡ 1 modulo the prime ideal above p in ZZ[ζ_{μ}].

Let C_{q} denote the full subcategory of finite dimensional Q-algebras whose simple components are isomorphic to some of the cyclotomic fields Q[ζ_{μ}]. Note that G acts naturally on all objects of C_{q}. Consider a functor V from C_{q} to finite free Z-modules, which takes Q to 0 and commutes with direct products. Then Proposition 1.2 of [6] says that the cokernel of the natural map

α_{V} : ∏_{C} V ( QC) ⟶ V ( QA),

as C runs over all cyclic subgroups of A, is a finite p-group — provided that V satisfies the following condition: for every p-subgroup Γ ⊂ G, the Γ-fixed submodule V ( Q(ζ))^{Γ} is the isomorphic image of V ( Q(ζ^{|Γ|})) under the natural inclusion. Our first lemma checks this for V ( Q(ζ_{μ})) = Ω^{1}(ζ_{μ}).

Lemma 3.1: Let Γ ⊂ G be a p-group and put η = ζ^{|Γ|}. Then u ∈ Ω^{1}(ζ) is fixed under Γ if and only if u ∈ Ω^{1}(η). Proof: The prime ideal above p in ZZ[η] is the Γ-fixed part of the analogous ideal in ZZ[ζ]. Hence the lemma is just saying that Ω(η) = Ω(ζ) ∩ Q(η), a well-known fact, which we prove here for the sake of completeness.

First note that Ω(η) = N_{Γ}Ω(ζ) because η = N_{Γ}ζ and 1 − η = N_{Γ}(1 − ζ), as in Lemma 2.1. To see that Ω(ζ)^{Γ} ⊆ N_{Γ}Ω(ζ), remember that Ω(ζ) is monogenic over ZZ(G_{+}), indeed free modulo t_{+}, the sum over all elements of G_{+}, (cf. proof of 2.3). Now let u ∈ Ω(ζ) and u^{τ} = u, for all τ ∈ Γ. Then u = u_{0}^{β}, with β = ∑ b_{γ}γ ∈ ZZ(G_{+}), implies that τβ = β + kt_{+}, whence τβ = β by comparing augmentations. Therefore b_{γ} is constant on cosets of Γ, and β is a multiple of the trace over Γ.

Lemma 3.2: The cokernel of

α_{Ω} : ∏_{C} Ω(C) ⟶ Ω(A),

as C ranges over all cycles in A, is a finite p-group, which is trivial if p is regular. Proof: The multiplicative group E_{p}(A) = 1 + Δ_{p}(A), where Δ_{p}(A) denotes the kernel of the p-adic augmentation ZZ_{p}A → ZZ_{p}, is of finite index in E_{p}(M_{A}) = 1 + Δ_{p}(M_{A}), where Δ_{p}(M_{A}) = Δ_{p}(A)M_{A} is the direct sum of the appropriate prime ideals in the Wedderburn components. Since E_{p}(M_{A}) is a ZZ_{p}-module by [11], II.3, Proposition 9, this index is a p-power. For Ω^{1}(M_{A}) = Ω(M_{A}) ∩ E_{p}(M_{A}), this implies that Ω^{1}(M_{A})/Ω(A) is a (finite) p-group. In fact, for every u ∈ Ω^{1}(M_{A}) a suitable p-power v lies in E_{p}(A). Thus v has rational coefficients that are integral with respect to p. On the other hand, |A|v ∈ ZZA because v ∈ M_{A} (cf. [9], Theorem 41.1) and therefore v ∈ ZZA ∩ Ω(M_{A}) = Ω(A).

Now look at the commutative square

∏_{C} Ω(C) ⟶ Ω(A)
↓  ↓
∏_{C} Ω^{1}(M_{C}) ⟶ Ω^{1}(M_{A}).

We have just seen that all [Ω^{1}(M_{C}) : Ω(C)] are p-powers. Since Lemma 3.1 permits us to invoke the quoted result from [6], the bottom horizontal arrow has p-power index, too.

Since the cokernel in question is a p-group, it can also be obtained from the map

αΩ̂ : ∏_{C} Ω̂(C) → Ω̂(A),

where the “hat” denotes p-adic completion (tensoring with ZZ_{p}) as before. Now, if p is regular, Remark 2.7 shows that Ω̂(A) = Ŝ(A), so that αΩ̂ shows up as the northern horizontal in the diagram

∏_{C} Ŝ(C) ⟶ Ŝ(A)
↓  ↓
∏_{C} S′_{p}(C) ⟶ S′_{p}(A),

whose southern horizontal is surjective by [4]. The proof is finished by remarking that, for regular p, the verticals are bijective, as has been proved in [7].

Corollary 3.3: The natural map

λ_{A} : ∏_{C} W (C) ⟶ Ω(A),

with the product direct and C ranging over all cycles in A, is an injection of p-power index; it is bijective if p is regular. Proof: For cyclic A this is just a summary of the last section from Lemma 2.6 onward. Thus, for every u_{C} ∈ Ω(C) a certain p-power is constructible, i.e. in Λ(C). By Lemma 3.2 above, for every u ∈ Ω(A) a certain p-power is a product of such u_{C}, hence a possibly higher p-power will be constructible. These p-powers are trivial in the regular case.

To see that λ_{A} is injective, we note that the direct product ∏_{C} W (C) has the same rank as Ω(A), which is the same as that of Ω(M_{A}). Indeed, section 1 shows that L(η) = Ω(η)^{σ−1}, where η is a |C|-th root of 1, and σ generates G. Since u ↦ u^{σ−1} is injective on S(C), this implies that L(η), and hence W (C), has the same rank as Ω(η). Since the number and size of cyclic subgroups of A are in one-to-one correspondence with those of cyclic factor-groups, the rank of Ω(M_{A}) equals the sum of the ranks of the W (C) as C ranges over cycles in A, as was to be shown.

Remark 3.4: Thus the group Λ(A) = im(λ_{A}) of constructible units is the direct product of the groups W (C). In particular, if B ⊆ A is a subgroup, it follows that

Λ(A) = Λ(B) × ∏_{C′⊈B} W (C′).  (10)

This implies that the notion of constructiblity does not depend on the ambient group. In other words, a unit u ∈ Ω(B) cannot be made constructible by going to a larger group A; in symbols:

Λ(A) ∩ Ω(B) = Λ(B).  (11)

In fact, if u ∈ Λ(A) ∩ Ω(B), a certain power u^{N} lies in Λ(B). Now, if we factor u = v · v′ according to (10), the N-th power of v′ would be trivial. But there is no torsion, hence v′ = 1.

If p is irregular, Lemma 2.9 shows that there are non-constructible units in Ω(C) for every cyclic C ⊆ A with |C| > p. By (11) they persist non-constructibly in Ω(A). Let us summarize.

Theorem 3.5: Take A of order > p, and let w(−) be the polynomial described in the introduction. Then Ω(A) contains the direct product

Λ(A) = ∏_{x} w(x)^{ZZG}

as a subgroup of p-power index c(A), where x ∈ A runs over a complete system of representatives of G-orbits; and c(A) = 1 if and only if p is regular.

Proof: Almost everything follows immediately from 3.3 and 3.4, except for one small item: it remains to be shown that Λ(A) ≠ Ω(A) if p is irregular and A is non-cyclic, elementary abelian (i.e. m = 1). This case, obviously beyond the reach of Lemma 2.9, requires special treatment, which is almost contained in [3], but not quite. Unfortunately [3] is geared to S(A) instead of Ω(A), and its results, stated only for semi-regular p, cannot be used here without modification.

We shall sketch an adaptation of the arguments of [3], referring to that paper for details. Let A^{p} = 1, but |A| > p.

Consider the cokernel M of the injection

∏_{K} Ω(K)/Ω(A) ⟶ ∏_{K} S′_{p}(K)/S′_{p}(A),  (12)

where K runs over the non-trivial cyclic factor-groups of A. Remember that Ω(K) = W (K) in this case. We contend that

(i) if p is irregular, M is non-trivial; (ii) if Λ(A) = Ω(A), it is trivial.

As to (i), M can also be identified as the cokernel of the map

δ : ∏_{K} Ω̂(K) × S′_{p}(A) ⟶ ∏_{K} S′_{p}(K),

which is the “difference” of the obvious two inclusions. For each of the relevant Bernoulli numbers which vanishes mod p, the index of the first of these goes up by an amount that cannot be compensated by the second one. The numbers involved are exactly as in [3], p.260.

As for (ii), first note that [∏_{K} Ω(K) : ∏_{C} W (C)] = [∏_{K} S′_{p}(K) : ∏_{C} S′_{p}(C)]. In fact, all four ZZ_{p}G_{+}-modules mentioned here are direct powers of one and the same ZZ_{p}G_{+}/(t_{+}) as in Theorem 2.3 above. To show this in the p-adic case, we use the logarithm (cf. [3], p.256) on S_{p}(C), and see that S′_{p}(C) is generated over ZZ_{p}G_{+} by exp(x^{σ} − x + x^{−σ} − x^{−1}). Now, if we translate the maps ∏ W (C) → ∏ Ω(K) and ∏ S′_{p}(C) → ∏ S′_{p}(K) into matrices over ZZ_{p}G_{+}/(t_{+}), we actually find the same matrix in each case – cf. [3], Proposition 3.1.

Finally ∏ S′_{p}(C) = S′_{p}(A) for every odd p by [4], and if we also happen to have ∏ W (C) = Ω(A), it now follows that the two groups appearing in (12) have equal orders, i.e. M is trivial. This establishes (ii) and finishes the proof.

Remarks 3.6:

1. We have chosen the notation c(A) to suggest the term circular index for [Ω(A) : Λ(A)], in distinction to the previously studied Bass-Milnor index b(A), which measures the cokernel of the map

α_{S} : ∏_{C} S(C) ⟶ S(A).

If m = 1 we obviously have W (C) = Λ(C) = Ω(C) for all cyclic C, and hence c(A) equals the index of the map αΩ̂ mentioned in Lemma 3.2. If p is semi-regular (cf. Remark 2.7), this is clearly the same as b(A). However, when m > 1 these indices no longer agree. For instance, if A is cyclic of order p^{2} and p is irregular, c(A) ≠ 1 by 2.9, but obviously b(A) = 1. The diagram

∏_{C} Ŵ (C) ⟶ Ω̂(A)
↓  ↓
∏_{C} Ŝ(C) ⟶ Ŝ(A)

shows that b(A) always divides c(A), at least if p is semi-regular.

Under the assumption of semi-regularity and for elementary abelian A, the index b(A) was found to be an essential factor in the class-number of ZZA (again cf.[3]). More precisely, it divides the order of the part D(A) of the ideal class group which “capitulates” in the transition ZZA → M_{A}.

This cannot be generalized directly to non-elementary A, because b(A) does not adequately reflect the non-triviality of D(A) for A cyclic of order > p. Strangely enough it is the circular index c(A) which appears to play the appropriate role, thus acquiring an unexpectedly intrinsic status. As we saw in the last proof, it even seems to be the better invariant in the elementary abelian case, where, however, the assumption of semiregularity confuses it with b(A). A precise statement of the connections between D(A) and c(A) would require the deployment of concepts and notation well outside the present context. Suffice it to say that they generalize the findings of [3] with c(A) supplanting b(A), that they have been verified for several classes of A, and that they will be published separately as soon as a proof of the general case becomes available.

2. To finish, we exhibit the explicit form of the basic constructible unit w_{g,h}(x) for the case that 2 is a primitive root modulo p. For x ∈ A and any natural number ν, consider the element

δ_{ν}(x) = ∑_{i=1}^{ν−1}(x^{i} + x^{−i}) − (x^{i+ν} + x^{−i−ν})

of ZZA. Taking g = 2 and h = (q + 1)/2, we then have the following two forms, depending on the class modulo 4 of q:

w_{g,h}(x) = \{ \begin{matrix}−1 − δ_{t}(x) + x^{2t} + x^{−2t}, & if q = +1 + 4t; \\ +1 + δ_{s}(x), & if q = −1 + 4s. \end{matrix}

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