GALOIS STABILITY FOR INTEGRAL REPRESENTATIONS OF FINITE ABELIAN GROUPS D.A.Malinin

ABSTRACT For a given field F of characteristic 0 we consider a normal extension E/F of finite degree d

and finite abelian subgroups G⊂GL_{n}(E) of a given exponent t. We assume that G is stable under the natural action of the Galois group of E/F and consider fields E=F (G) that are obtained via adjoining all matrix coefficients of all matrices g∈G to F. It is proved that under some reasonable restrictions for n any E can be realized as F (G), while if all coefficients of matrices in G are algebraic integers there are only finitely many fields E=F (G) for prescribed integers n and t or prescribed n and d.

1. Introduction

In this paper we consider some Galois extension E/F of finite degree d with the Galois group Γ for a field F of characteristic 0 and a finite abelian subgroup G ⊂ GL_{n}(E) of the given exponent t, where we assume that G is stable under the natural coefficientwise Γ-action.

Throughout this paper O_{E} is the maximal order of E and F (G) denotes a field that is obtained via adjoining to F all matrix coefficients of all matrices g ∈ G.

The main objective of this paper is to prove the existence of abelian Γ-stable subgroups G such that F (G) = E provided some reasonable restrictions for the fixed normal extension E/F and integers n, t, d hold and to study the interplay between the existence of Γ-stable groups G over algebraic number fields and over their rings of integers.

The results related to the Galois stability of finite groups in the situation similar to ours arise in the theory of definite quadratic forms and Galois cohomologies of certain arithmetic groups if F is an algebraic number field and G is realized over its maximal order ([B], see also [Ro]). In our context we study whether a given field E normal over F can be realized as a field E = F (G) in both cases G ⊂ GL_{n}(E) and G ⊂ GL_{n}(O_{E}), and if this is so what are the possible dimensions n of matrix realizations and the structure of G. Some similar questions for Γ-stable orders in simple algebras are considered in [RW], see also [RW1] for some applications.

We give a positive answer to the first question: we prove that any finite normal field extension E/F can be obtained as F (G)/F if n ≥ ϕ_{E}(t)d where ϕ_{E}(t) = [E(ζ_{t}) : E] is the generalized Euler function and ζ_{t} is a primitive t-root of 1. An explicit construction of these fields is given in Theorem 1 in section 2. In fact, we construct some Galois algebras in the sense of [ILF], and we establish the lower bounds for their possible dimensions n. We show (see Proposition 1 in section 2) that the restrictions for the given integers n, t, and d in Theorem 1 can not be improved. We also study the problem of uniqueness of possible realizations for cyclic, abelian and solvable extensions E/F provided the ground field F contains roots of 1 of an appropriate degree (Theorems 2 and 3 in section 2).

The author had a nice opportunity to work on this subject during his stay at the MPI and would like to express his gratitude for hospitality. The author is grateful to DAAD for support.

The situation becomes different if E is an algebraic number field and all matrix coefficients of g ∈ G are algebraic integers. The existence of any Galois stable subgroups G ⊂ GL_{n}(O_{E}) such that F (G) ≠ F is a rather subtle question. In particular, for F = ℚ all fields F (G) whose discriminant is divisible by an odd prime must contain non-trivial roots of 1 [BM], [M1].

We determine certain properties of inegral realizations for the groups stable under the Galois action in section 3, and we consider the question, whether the given field E is realizable as E = F (G) for a Γ-stable G ⊂ GL_{n}(O_{E}) . In this situation the typical answer is negative (we discuss it in section 3.1, in particular, see Theorem 4), but some exeptions are possible, specially if some nontrivial roots of 1 are contained in the field E.

The paper continues the research of [M5] and uses the approach from [M1] and [M3] based on the integral representation theory of finite groups. It is organized in the following way. We describe certain Γ-stable groups G over fields and give the statement of the results in section 2 and their proofs in section 4. Section 3 is completely devoted to integral (in O_{E} and some other subrings of E) representations of finite groups that are stable under Γ-action. Some of results in this section are valid without restriction for G to be abelian (Theorem 4, Theorem 6, 2), Lemma 1). Section 3 is subdivided in three parts. Some preliminaries are given in 3.0. In 3.1 we prove that ”almost all” fields in certain probabilistic sense are not realizable as F (G) (Theorem 4). We also show that there is only a finite number of possible fields F (G) if F, [F (G) : F] and n are fixed, or F, n and t are fixed provided G is abelian of exponent t (Theorem 6) and discuss some related questions. In 3.2 we show the way to apply Theorem 1 to the problem of existence of integral Γ-stable representations. In the end of 3.2 we give an example of a non-maximal order R, O_{F} ⊂ R ⊂ O_{E}, where F contains a primitive t-root of 1, such that any Γ-stable subgroup G ⊂ GL_{n}(R) of the exponent prime to t is already contained in GL_{n}(O_{F}) where O_{F} is the maximal order of F.

Notation

Most of symbols and notations that we use in this paper are traditional. ℚ is the field of rationals. GL_{n}(R) denotes the general linear group over R. [E : F] denotes the degree of the field extension E/F. We write ζ_{t} for a primitive t-root of 1. We denote by ϕ_{K}(t) = [K(ζ_{t}) : K] the generalized Euler function for a field K. I_{m} stands for the unit m × m-matrix, 0_{n,m} and 0_{m} are zero n × m and m × m-matrices, e_{i,j} are square matrices having the only nonzero element 1 in the position (i, j), detM is the determinant of a matrix M, diag(d_{1}, d_{2}, ..., d_{m}) is a diagonal matrix having diagonal components d_{1}, d_{2}, ..., d_{n}. If G is a finite linear group, F (G) stands for a field obtained by adjoining to F all matrix coefficients of all matrices g ∈ G. The symbol Γ is used to denote Galois groups. For Γ acting on G and any σ ∈ Γ and g ∈ G we write g^{σ} for the image of g under σ-action. dim_{K}A denotes the dimension of K-algebra A over the field K.

2. Galois stability for representations over fields

Firstly we are interested in the following existence theorem:

Theorem 1. Let F be a field of characteristic 0, let d > 1, t > 1 and n ≥ ϕ_{E}(t)d be given integers, and let E be a given normal extension of F having the Galois group Γ and degree d. Then there is an abelian Γ- stable subgroup G ⊂ GL_{n}(E) of the exponent t such that E = F (G).

Note that the dimension n = dϕ_{E}(t) in our construction is the minimum possible.

As a corollary of Theorem 1 we have

Proposition 1. Let E/F be a given normal extension of algebraic number fields with the Galois group Γ, [E : F] = d, and let G ⊂ GL_{n}(E) be a finite abelian Γ-stable subgroup of exponent t such that E = F (G) and n is the minimum possible. Then n = dϕ_{E}(t) and G is irreducible under conjugation in GL_{n}(F). Moreover, if G has the minimum possible order, then G is a group of type (t, t, ..., t) and order t^{d}.

In the case of quadratic extensions we can give an obvious example.

Example. Let d = 2, t = 2. Pick E = ℚ(\sqrt{a}) and g = | \begin{matrix}0 & 1 \\ a^{−1} & 0 \end{matrix} | \sqrt{a} for

any a ∈ F which is not a square in F. Then Γ is a group of order 2 and G = {I_{2}, −I_{2}, g, −g} is a Γ-stable abelian group of exponent 2.

If the ground field F contains roots of 1 of sufficiently large degrees then the construction and the description of finite abelian Γ-stable groups with coefficients in extended fields can be simplified. We shall give an example of such description in the case of a cyclic Kummer extension of fields E/F of degree q which has Galois group Γ generated by a single element σ. We also assume that F contains primitive roots ζ = ζ_{t} and ζ_{q} of degrees t and q respectively. We assume that some element π ∈ E generates a Kummer basis 1, π, π^{2}, ..., π^{q−1} so that π^{q} ∈ F.

Let q_{i} be some divisor of q, and let r_{i} = \frac{q}{q_{i}}. Set

B = | \begin{matrix}0 & 1 & 0 … & 0 \\ 0 & 0 & 1 … & 0 \\ . & . & . \\ & & & 1 \\ π^{−q} & 0 & 0 … & 0 \end{matrix} | ,

the matrix of regular representation of E_{1}, the subfield of σ^{q_{i}}-fixed elements of E, over F in the basis 1, π^{r_{i}}, π^{2r_{i}}, ..., π^{(q−1)r_{i}} of E/F.

Let us consider the following matrices:

g = I_{q_{i}} + \frac{1}{q_{i}} ∑_{j=0}^{q_{i}−1} π^{jr_{i}}(ζ − 1)B^{j}; g_{1} = I_{m} ⊗ g.

Then the group G_{m,q_{i}} of matrices which is generated by all elements g_{1}^{σ^{i}}, i = 1, 2, …, q − 1, is a finite abelian Γ-stable subgroup of GL_{mq_{i}}(E) of exponent t. We intend to describe all Γ-stable subgroups H of exponent t in GL_{n}(E) for some positive integer n.

Theorem 2. Every finite abelian Γ-stable subgroup H ⊂ GL_{t}(E) of exponent t is conjugate in GL_{n}(E) to a subgroup H_{1} ⊂ ⊕_{k}G^{k}, where G^{i} ⊂ G_{m,q_{i}}, or G^{k} consists of diagonal matrices.

One application of Theorem 2 is given as an example in the end of section 3.1.

A similar description of finite Γ-stable abelian subgroups of exponent t can be given in the case of finite abelian and solvable extensions. For simplicity we consider the abelian case.

Now, we can extend the result of Theorem 2 to the case of abelian extension E/F provided F contains a primitive roots ζ = ζ_{t} and ζ_{l} of 1, where l is the exponent of the Galois group Γ of E/F.

Let Γ be a direct sum of cyclic groups whose orders are q_{1}, q_{2}, ..., q_{v}. Since E is a composite of cyclic Kummer extensions E_{i}/F having degree n_{i}, we can fix a Kummer basis 1, π_{i}, π_{i}^{2}, ..., π_{i}^{q_{i}−1} for each of them.

The dimension of our matrix realization is n = mq′_{1}...q′_{k} for some divisors q′_{i} of q_{i}, k ≤ v. We write q = q_{1}q_{2}...q_{k}, r_{i} = \frac{q_{i}}{q′_{i}} and r for the set (r_{1}, r_{2}, ..., r_{k}).

Set B_{0} = I_{mq_{i}},

B_{i} = I_{m} ⊗ | \begin{matrix}0 & 1 & 0 … & 0 \\ 0 & 0 & 1 … & 0 \\ . & . & . \\ & & & 1 \\ π_{i}^{−q_{i}} & 0 & 0 … & 0 \end{matrix} |

(all nonzero elements of B_{i} are: 1 in positions (12), (23), ...(q′_{i} − 1, q′_{i}) and π_{i}^{−q_{i}} in the position (q′_{i}1) ).

Let us construct matrices B_{I} inductively for indices I = (i_{1}, i_{2}, ..., i_{k}), k ≤ v, 0 ≤ i_{j} < n_{j}. Let

B_{I} = | \begin{matrix}0_{p} & I_{p} & 0_{p} … & 0_{p} \\ 0_{p} & 0_{p} & I_{p} … & 0_{p} \\ . & . & . \\ & & & I_{p} \\ π_{i}^{−q_{i}}B_{I′} & 0_{p} & 0_{p} … & 0_{p} \end{matrix} |

for p = q′_{1}q′_{2}...q′_{k−1}, I′ = (i_{1}, i_{2}, ..., i_{k−1}) and pq_{i}×pq_{i}-matrices B(i_{1}, i_{2}, ..., i_{k−1}, 0) = diag(B_{I′}, ..., B_{I′}), where B_{I′} is a p × p-matrix, i_{j} = 0, 1, ..., n_{j} − 1

Define a matrix as follows:

g = \frac{1}{q} ∑_{(i_{1}...i_{k})}((ζ_{t} − 1)π_{1}^{i_{1}r_{1}}π_{2}^{i_{2}r_{2}}...π_{k}^{i_{k}r_{k}}B_{(i_{1},i_{2},...,i_{k})} + I_{n}).

Let G_{m,k,r} be the group generated by all matrices g^{σ}, σ ∈ Γ.

Theorem 3. Any finite Γ-stable subgroup G ⊂ GL_{n}(E) of exponent t is conjugate in GL_{n}(F) to a subgroup H of the direct sum ⊕_{i}G^{(i)}, where G^{(i)} is contained in either G_{m,k,r} for some integers m, k(k ≤ v) and some r, or a group of diagonal matrices.

The matrices B_{I} can be transformed to a different simplified form, see Note in the end of the proof of Theorem 3.

The proof of the results of this section will be given in section 4.

3. Galois stability for integral representations

3.0. Preliminaries.

In this section we introduce representations over the maximal order O_{E} of an algebraic number field E. Therefore, we should distinguish some special properties of the realization fields for Galois stable groups over algebraic number fields and their maximal orders. Integral representations in question are very sensitive to the changing of the ground ring, the ramification index, and the group of units of O_{E}, specially its torsion subgroup, play an important role ([M1], [M2], [M3]). As it was mentioned in the introduction, the existence of Galois stable subgroups G ⊂ GL_{n}(O_{E}) such that F (G) ≠ F is a rather subtle problem since the Galois stability of G imposes strong restrictions for F (G) such that in the case F = ℚ all fields F (G) whose discriminant is divisible by an odd prime must contain some cyclotomic subfield K ≠ ℚ [BM], [M1]. The question, whether the given field E is realizable as E = F (G) for a Γ-stable G, can be formulated for subgroups G ⊂ GL_{n}(R) for certain ring extensions R ⊃ R_{0} with quotient fields E and F respectively. In section 3.2 we give an example for the property of non-realizability of the Galois stable matrix groups over certain non-maximal orders in Kummer extensions which can be also compared to the existence of appropriate realizations for abelian groups over fields given in Theorems 2 and 3.

There are numerous classical results related to the description of integral representations of finite abelian groups over Dedekind rings R, specially for the rings of integers ℤ or p-adic integers ℤ_{p} and maximal orders of the local fields and the fields of algebraic numbers. Many of them go back to the classification of indecomposable representations that can give an explicit description only for narrow classes of groups rings R (see [R] for a survey and [D] for the classification of finite, wild and tame representation types). The class of matrix groups that are studied in this paper is much more narrow and special since we assume their stability under the natural Galois group action. It is also possible to reduce some general problems concerning these groups to the case of indecomposable or irreducible Galois stable representations. In [M1], [M4] the author has introduced the class C(F) of fields normal over F which are obtained by adjoining to F all coefficients of matrices contained in some finite Γ-stable group G ⊂ GL_{n}(O_{E}) and implemented the theory of integral representations of finite groups to study C(F) first of all in the case F = ℚ. The results of this section emanate from the approach of [M1, M4] in the spirit of [M5] and give an extension of results that have been obtained in these papers.

Besides a score of generalizations, finite groups that are stable under Galois action have some interaction with seemingly unrelated results in the theory of definite quadratic forms and Galois cohomologies of certain arithmetic groups, if F is an algebraic number field and G is realized over its maximal order [B], [BK].

Since in our context we are interested in the existence and integral realizability of finite Γ-stable groups, it is convenient to study elementary abelian subgroups of the type (t, t, ..., t) (comp. Proposition 1 in section 2). If F is an algebraic number field, it is proved in [M], [M1] that the general case can be reduced to our situation, more precisely, the existence of Γ-stable groups G ⊂ GL_{n}(F) such that F (G) ≠ F (for some F, in particular F = ℚ) implies the existence of abelian Γ-stable groups of exponent p having the same property for some ramified prime p.

These realizations appear to be integral in some special cases (we discuss the case of some cyclic extensions E/F in the end of 3.1) or can be realized integrally (see section 3.2).

Our approach to solve the problem of description of all Γ-stable matrix groups up to GL_{n}(R)-conjugation for certain Dedekind rings R ⊂ E is based on the criterion in 3.2 (Theorem 7) for the existence of integral realization of the given Γ-stable subgroup G ⊂ GL_{n}(E). So, if we have a description of G up to GL_{n}(F)-conjugation, we can also determine whether G is GL_{n}(F)-conjugate to a subgroup of GL_{n}(R) for any fixed n, E and F. In fact, we have an algorithm to answer the question: for a given field extension E/F is it possible to find a Γ-stable subgroup G ⊂ GL_{n}(R) which is not contained in GL_{n}(F)?

3.1. Quantative results and finiteness theorems.

Let O_{K} and O_{E} denote the maximal orders of number fields K and E respectively. The following result establishes ”rarity” of possible realizations in the sense of probability to find the given field as a splitting field of a certain polynomial.

Let us suppose that K = ℚ, the field of rationals, or K = ℚ(\sqrt{d}), d < 0, d ∈ ℤ. We consider the set O(N) = {α ∈ O_{K} | |N_{K/ℚ}(α)| ≤ N} where N_{K/ℚ} is the norm map. The proof of the following Theorem is based on the result by S. D. Cohen (Theorem 1 in [C]) combined with some asymptotic estimates for the number of integral polynomials having bounded coefficients with respect to the norm and reducible over K(\sqrt{b}) (b is contained in a finite set of elements from O_{K}):

Theorem 4. Let v(N) denote the total number of polynomials of degree m with coefficients in O(N), and let ψ(N) denote the number of those polynomials whose splitting fields do not contain any fields K(G) ≠ K for G ⊂ GL_{n}(O_{E}), E ⊃ K and fixed n. Then

lim_{N→∞} \frac{ψ(N)}{v(N)} = 1.

Theorem 4 shows that ”almost all” fields are not realizable via adjoining matrix coefficients of Γ-stable groups to the field of rational numbers or its imaginary quadratic extensions if this coefficients are contained in the rings of integers of algebraic number fields.

On the contrary, according to Theorem 1, all algebraic number fields are realizable in our situation if the matrix coefficients are admitted to be contained in the field, not its maximal order.

Proof of Theorem 4. We use properties of distribution of Galois groups of polynomials that were considered by S. D. Cohen [C]. In accordance with [C] the number of polynomials in question having the symmetric Galois group S_{m}, divided by the total number of polynomials in question, tends to 1 when N → ∞. Therefore, we can consider only the number of these K-irreducible polynomials which are reducible over K(\sqrt{α}) for a finite number of α. The elements \sqrt{α} can be contained only in a finite number of extensions K(G) which have no ramified primes p ⩾ m! + 1 and have degree m! over K. Let us estimate the number of these polynomials. Letf ∈ O(N)[x] and f = g · g′, g, g′ ∈ K(\sqrt{α})[x]. Let E ∈ O_{K(\sqrt{α})} be a unit of infinite order. We can suppose that after some adjustment both height |g| = max |a_{i}| of g = ∑ a_{i}x^{i} and height |g′| of g′ = ∑ a′_{i}x^{i} are equal up to a constant c = c(K, m). Indeed, let |g| = A, |g′| = B, |f| = c_{0}N, c_{0} = c_{0}(K, m). Let t = log_{E} (\frac{A}{\sqrt{N}} ), then changing g and g′ to p = E^{−[t]}g and p′ = E^{[t]}g respectively we obtain |p| ∼ \sqrt{N}, |p′| ∼ \sqrt{N}, that is |p| ⩽ c_{1}(K, m)\sqrt{N} and |p′| ⩽ c_{2}(K, m)\sqrt{N}. As p = p_{1} + \sqrt{α}p_{2} and p′ = p′_{1} + \sqrt{α}p_{2} for p_{i}, p′_{i} ∈ K[x] and p′ = p^{σ} for nonidentical automorphism σ of K(\sqrt{α}) over K, we can see that |p_{i}| ⩽ c_{3}\sqrt{N} and |p′_{i}| ⩽ c_{3}\sqrt{N} for i = 1, 2 and c_{3} = c_{3}(K, m). Therefore, there are only (c_{2}\sqrt{N})^{2·(m/2+1)} = c_{4}N^{m+2}, c_{4} = c_{4}(K, m), polynomials that are reducible over K(\sqrt{α}). Likewise, there are c_{5}N^{2(m+1)}, c_{5} = c_{5}(K, m), polynomials f in O(N)[x] and it is obvious that

lim_{N→∞} \frac{c_{4}N^{m+2}}{c_{5}N^{2m+2}} = 0.

So, we showed that the number of polynomials whose splitting fields can contain any K(G) ≠ K is small asymptotically, and our estimate completes the proof of Theorem 4.

Let E/F be a given normal extension of algebraic number fields. Then we write Γ_{E} for the Galois group of E over F. Let t > 0 be a given integer. Let ζ_{t} be a primitive root of 1 of degree t. We use notation γ_{t} = [E(ζ_{t}): F (ζ_{t})].

Theorem 5. Let a Γ_{E}-stable abelian subgroup G ⊂ GL_{n}(E) of exponent t be irreducible under GL_{n}(F)-conjugation, and let E = F (G). Then γ_{t} divides n.

Proof of Theorem 5. Let w_{1}, w_{2}, …, w_{r} be some basis of E(ζ_{t}) over F (ζ_{t}). Let G be reducible under conjugation in GL_{n}(F (ζ_{t})). Then G splits into components of equal dimensions, each of them being GL_{n}(F (ζ_{t}))-irreducible. Therefore, we can restrict ourselves to considering a GL_{n}(F (ζ_{t}))-irreducible component and use the same notation G for it. Let ε_{i} = ∑_{j} A_{ij}w_{j} be a primitive idempotent of E(ζ_{t})-algebra E(ζ_{t})G. We can prove that E(ζ_{t}) is obtained by adjoining to F (ζ_{t}) all eigenvalues of matrices A_{ij}. Indeed, this eigenvalues are equal to λ_{ij} = \frac{det W_{ij}}{det W} for W = [w_{i}^{σ_{j}}], where Γ = \{σ_{1}, σ_{2}, …, σ_{r}\} is the Galois group of E(ζ_{t})/F (ζ_{t}), and W_{ij} is obtained from W by replacing its j-th column with t(0 … 1 … 0) (i-th element is 1, all other elements are 0), and λ_{ij} are precisely elements of the matrix W^{−1}. Furthermore, if ∑ α_{ij}λ_{ij} = θ is a primitive element of E(ζ_{t}) over F (ζ_{t}) for some α_{ij} ∈ F, then the matrix M = ∑ α_{ij}A_{ij} ∈ F G and its spectrum consists of all conjugate elements θ^{σ}, σ ∈ Γ, in virtue of irreducibility of G. It follows that the number of eigenvalues of M is divisible by γ_{t}, and γ_{t} divides n. This completes the proof of our claim.

We note that the reduction to the case of an irreducible group G is motivated by the following lemma:

Lemma 1. If G ⊂ GL_{n}(E_{1}) is a finite Γ_{E}-stable subgroup which has GL_{n}(F_{1})- irreducible components G_{1}, G_{2}, …, G_{r}, and E_{1}, F_{1} are rings having quotient fields E and F respectively, then F (G) is the composite of fields F (G_{1}), F (G_{2}), …, F (G_{r}).

Proof of Lemma 1. Let

G^{1} = h^{−1}Gh ⊂ | \begin{matrix}G_{1} & & & & ∗ \\ & . & . & . \\ 0 & & & & G_{r} \end{matrix} |

for h ∈ GL_{n}(F_{1}). If there exists g ∈ G such that g^{γ} ≠ g for some automorphism γ of F (G) over F (G_{1})F (G_{2}) … F (G_{r}), then g_{1} = g^{γ}g^{−1} ≠ I_{n}. But all eigenvalues of g_{1} are 1. This contradiction completes the proof of Lemma 1.

Clifford’s theory of restriction of irreducible representations to the normal subgroups allows us to describe non-abelian subgroups G ⊂ GL_{n}(O_{E}).

We can give some quantative characterizations of groups G in question.

Theorem 6. 1) For a given number field F and integers n and t, there are only a finite number of normal extensions E/F such that E = F (G) and G is a finite abelian Γ_{E}-stable subgroup of GL_{n}(O_{E}) of exponent t.

2) For a given number field F and integers n and d = [E: F], there is only a finite number of fields E = F (G) for some finite Γ_{E}-stable subgroup G of GL_{n}(O_{E}).

But if we admit the ground ring to be the field E instead of O_{E} we can compare Theorem 6 and the following

Proposition 2. For a given number field F and given integers d > 1, t > 1 and n ⩾ [F (ζ_{t}): F] · d, there are infinitely many normal extensions E/F of degree d such that E = F (G) for some finite Γ_{E}-stable abelian subgroup G ⊂ GL_{n}(E) of exponent t.

Proposition 2 is an immediate corollary of Theorem 1.

Proof of Theorem 6. 1) In virtue of Lemma 1 we can restrict ourselves to considering only irreducible G. It follows from integrality in O_{E} of all coefficients of G that only divisors of t can ramify in E, and in accordance with Theorem 5 the degree [E: F] is restricted by a constant that depends only on t and n. Furthermore, it follows from the formula

d_{K/ℚ} = N_{K_{0}/ℚ}(d_{K/K_{0}})d_{K_{0}/ℚ}^{r}, r = [K: K_{0}]

for discriminants of the tower K ⊃ K_{0} ⊃ ℚ of number fields that there is only a finite number of unramified extensions of the given number field of the prescribed degree. Since the number of algebraic number fields having the prescribeded discriminant is finite, and the power of the given ramified prime p that divides the discriminant of number field having the prescribed degree is restricted, we can obtain only a finite number of possibilities for the given n and t. Therefore, we have only a finite number of fields E that satisfy our conditions.

2) We can note that if prime p is ramified in E, then \frac{d}{p−1} ⩾ 1, that is p ⩽ d + 1. Indeed, Bartels proved in [B] that ramification index e of p in this situation satisfies inequality e ⩾ p − 1, and it is clear that d = [E : F] is always not less than e. Now we can use the proof given in 1).

Example. If the ground field F contains roots of 1 of sufficiently large degrees the description of finite abelian Γ_{E}-stable groups can be simplified. Here we give an example in the case of a cyclic ramified extension of the number fields E/F of a prime degree q whose Galois group Γ_{E} is generated by an element σ. Let us suppose that all prime ideals 𝔮 of O_{E} that do not divide a prime p ≠ q are unramified in E/F and the field F contains some primitive roots of 1 ζ and ζ_{q} of degrees p and q respectively. Moreover, we assume that there exists an element π ∈ O_{F} which is an uniformizing element in all ramified ideals of E, π^{q} ∈ F and the norm N_{E/F} (π) divides p. Set b = \frac{1}{q}(1 + π + π^{2} + ⋯ + π^{q−1}). Let (ζ − 1)B be a matrix of the regular representation of an element (ζ − 1)π^{−1} in the ring O_{F} [b, b^{σ}, …, b^{σ^{q−1}}]. Let us consider the following matrices:

g = I_{q} + \frac{1}{q} ∑_{i=0}^{q−1} π^{i}(ζ − 1)B^{i}; g_{1} = I_{m} ⊗ g.

It is not difficult to prove that the group G_{m} of matrices which is generated by all elements g_{1}^{σ^{i}}, i = 1, 2, …, q − 1, is a finite abelian Γ_{E}-stable subgroup of GL_{mq}(O_{E}) of exponent p.

We can use Theorem 2 to state that every finite abelian Γ_{E}-stable subgroup H ⊂ GL_{t}(E) of exponent p is conjugate in GL_{t}(E) to a subgroup H_{1} ⊂ ⊕G^{i}, where either G^{i} ⊂ G_{m} or G^{i} consists of diagonal matrices.

3.2. On the existence of integral representations stable under Galois action.

Let us formulate a criterion for the existence of an integral realization of an abelian group G with properties introduced above.

Let E, L be finite extensions of a number field F. Let O′_{E}, O′_{F}, O′_{L} be semilocal rings that are obtained by intersection of valuation rings of all ramified prime ideals in the rings O_{E}, O_{F}, O_{L}. Let w_{1}, w_{2}, …, w_{d} be a basis of O′_{E} over O′_{F}, and let D be the discriminant of this basis. Let us suppose that some matrix g of prime order p has coefficients in E and all Γ-conjugates g^{γ}, γ ∈ Γ generate a finite group G of exponent t. Let σ_{1} = 1, σ_{2}, …, σ_{d} denote all automorphisms of the Galois group Γ of E over F. Assume that L = E(ζ_{1}, ζ_{2}, …, ζ_{d}) where ζ_{1}, ζ_{2}, …, ζ_{d} are the eigenvalues of the matrix g. We shall reserve the same notations for certain fixed extensions of σ_{i} to L. Let E = F (G) be obtained by adjoining to F all coefficients of all g ∈ G. For an appropriate set of d eigenvalues ζ_{1}, ζ_{2}, …, ζ_{d} which depends on the primitive idempotents of algebra LG we the following Theorem is true:

Theorem 7. Let G be irreducible under GL_{n}(F)-conjugation. Then G is conjugate in GL_{n}(F) to a subgroup of GL_{n}(O′_{E}) if and only if all determinants

d_{k} = det | w_{1} … w_{k−1} ζ_{1} w_{k+1} … w_{d} ; w_{1}^{σ_{2}} … w_{k−1}^{σ_{2}} ζ_{2}^{σ_{2}} w_{k+1}^{σ_{2}} … w_{d}^{σ_{2}} ; ⋮ ; w_{1}^{σ_{d}} … w_{k−1}^{σ_{d}} ζ_{t}^{σ_{d}} w_{k+1}^{σ_{d}} … w_{d}^{σ_{d}} |

are divisible by \sqrt{D} in the ring O′_{L}.

Note that the conditions of Theorem 7 are always true if E is unramified over F since DO′_{E} = O′_{E} in this case.

The proof of Theorem 7 is constructive. It is based on the commutativity of the L-algebra LG and uses a system of linear equations that arises from simultaneous diagonalization of commuting matrices

g = ∑_{i=1}^{d} w_{i}B_{i} and g^{σ} = ∑_{i=1}^{d} w_{i}^{σ}B_{i}, σ ∈ Γ,

whose solutions are the eigenvalues of commuting matrices B_{i}, i = 1, 2, …, d.

Using some F -irreducible component of a semisimple algebra F G and concidering eigenvalues of B_{i} and g^{σ_{j}}, we can obtain the following system of linear equations in indeterminates x_{i}, i = 1, ..., d:

{ x_{1}w_{1} + x_{2}w_{2} + ⋯ + x_{t}w_{t} = ζ_{1}, ; x_{1}w_{1}^{σ_{2}} + x_{2}w_{2}^{σ_{2}} + ⋯ + x_{d}w_{t}^{σ_{2}} = ζ_{2}^{σ_{2}}, ; ⋮ ; x_{1}w_{1}^{σ_{d}} + x_{2}w_{2}^{σ_{d}} + ⋯ + x_{t}w_{t}^{σ_{d}} = ζ_{d}^{σ_{d}},  (∗)

The discriminant of this system is \sqrt{D}, where D = det[Tr_{E/F} (w_{i}w_{j})]_{ij} is the discriminant of basis w_{1}, w_{2}, …, w_{d}. The solutions of system (*) are x′_{i} = d_{k}(\sqrt{D})^{−1}. It follows from Γ-stability of the abelian group G that we can also obtain a system {λ_{i} = (x′_{i})^{σ_{j}}, j = 1, 2, …, t} of all different eigenvalues of the matrix B_{i}, Conversely, if we pick a system {λ_{i} = (x′_{i})^{σ_{j}}, j = 1, 2, …, d} of eigenvalues with an appropriate multiplicity and use the regular representation in the basis w_{1}, w_{2}, …, w_{d} of diagonal matrices whose diagonal elements are ordered in accordance with the action of automorphisms of L over F, we can obtain pairwise commuting matrices B_{i} and write g = w_{1}B_{1}+w_{2}B_{2}+⋯+w_{d}B_{d}. On the other hand, the values λ_{j}^{σ_{j}} are defined by λ_{j} and correspond uniquely to the eigenvalues of matrices g^{σ_{i}}, which are defined in a natural way, and g^{σ_{j}} commute with each other. Matrices g^{σ_{i}} generate a finite abelian Γ-stable group of exponent t.

We also use the fact that each semisimple matrix B ∈ GL_{n}(O′_{L}) is conjugate in GL_{n}(F) to a matrix from GL_{n}(O′_{F}) if and only if all its eigenvalues are contained in O′_{L}:

Lemma 2. 1) Let all eigenvalues λ_{i}, i = 1, 2, …, n of a semisimple matrix B ⊂ GL_{n}(F) be contained in the ring O′_{L} for some field L ⊃ F. Then B is conjugate in GL_{n}(F) to a matrix that is contained in GL_{n}(O′_{F}).

2) Conversely, if a matrix B are contained in GL_{n}(O′_{F}), then its eigenvalues are contained in O′_{L}.

Proof of Lemma 2. 1) If we consider the regular representation R : λ_{i} → GL_{d}(F) in the integral basis w_{1}, w_{2}, ..., w_{d}, then λ_{i} and all its Γ-conjugates are eigenvalues of R(λ_{i}) and all coefficients of R(λ_{i}) are contained in O′_{F}. Taking the direct sum of all irreducible components of these regular representations R with respect to all classes of Γ-conjugate eigenvalues and an appropriate multiplicity we obtain the representation R_{1} : B → GL_{n}(O′_{F}). The matrix B is conjugate to R_{1}(B) as we can see via reducing both semisimple matrices B and R_{1}(B) to the same rational normal form by conjugation in GL_{n}(F).

2) Consider the characteristic polynomial f(x) = a_{0} + a_{1}x + ... + a_{n}x^{n} of the matrix B. Since a_{n} = 1 and all a_{i} are in O′_{F} all roots of f(x) are in O′_{E}. This completes the proof of the Lemma 2.

The structure of semisimple matrices B_{i}, i = 1, 2, …, d and their pairwise commutativity implies that they are conjugate to integral (in O′_{L}) matrices simultaneously if all eigenvalues are in O′_{L}.

Applying this to matrices B_{i} we can obtain the proof of theorem 7.

Remark. The previous reasons remain true for other Dedekind subrings R ⊂ L. In particular, Theorem 7 can be proved for any semilocal rings R_{E}, R_{F} and R_{L} that are contained in O′_{E}, O′_{F} and O′_{L} respectively and can also be modified for the rings of integers O_{E}, O_{F} and O_{L} .

The question whether Γ-stability of a finite group G of matrices having coefficients in an extended ring R ⊃ R_{0} (where R_{0} = R^{Γ} is a subring of Γ-fixed elements) implies that G ⊂ GL_{n}(R_{0}) has in general a negative answer. This depends strongly on the structure of the group of units of R. But the answer can be positive even in the case when R_{0} contains nontrivial t-roots of 1. Here we give an example for groups G of order prime to the integer t.

Let L/K be a cyclic ramified tame Kummer extension of algebraic number fields of degree t having the Galois group Γ which is generated by an element σ. Let O_{K} and O_{L} be the maximal orders of K and L, and let 𝔭 be some prime divisor of O_{L} which is ramified in L/K. So the primitive t-root ζ = ζ_{t} of 1 is contained in K, and there is a prime element π ∈ O_{L} of 𝔭 such that π^{σ} = πζ. Let R = O_{K}[1, π, π^{2}, ..., π^{t−1}], and let G ⊂ GL_{n}(R) be a Γ-stable finite subgroup whose order m is prime to t.

We can prove that G ⊂ GL_{n}(R_{0}) for R_{0} = O_{K}. For this purpose let us consider any g ∈ G and its expansion

g = B_{0} + πB_{1} + π^{2}B_{2} + ... + π^{t−1}B_{t−1},

where all n×n-matrices B_{i} have coefficients in O_{K}. If there is an element γ ∈ Γ that acts non-identically on g we can suppose that γ = σ^{r} and its order is q^{i} for some prime q (if needed we can replace γ by some its power that does not fix g). Then we have

g − g^{γ} = (1 − ζ^{r})πB_{1} + (1 − ζ^{2r})π^{2}B_{2} + ... + (1 − ζ^{(n−1)r})π^{t−1}B_{t−1}

Note that all elements 1 − ζ^{rj}, j = 1, ..., n − 1 divide the prime q. We can consider the following congruences

g ≡ g^{γ}(mod(1 − ζ^{r})), g^{−1}g^{γ} ≡ I_{n}(mod(1 − ζ^{r})).

Let h = g^{−1}g^{γ}. Then h = I_{n} + B for some n × n-matrix B such that B ≡ 0_{n}(mod(1 − ζ^{r})). In virtue of the well-known properties of matrices congruent to I_{n} (see [Min1], sect.1, [Min2], sect.1, [Min3], p.186, [B], Hilfssatz 1, [M3], lemma 1 and [M2]) this congruence implies that the order of h is some power of q. But this is impossible since the orders of Γ and G have no common divisors.

Therefore, the natural Γ-action on G is trivial, Γ fixes all g ∈ G, and G ⊂ GL_{n}(O_{K}).

4. Proof of theorems 1, 2, 3 and Proposition 1

Proof of Theorem 1. For a given basis w_{1}, w_{2}, ..., w_{n} of E/F we intend to construct a matrix g = [g_{ij}]_{i,j} = ∑_{i=1}^{d} B_{i}w_{i} and pairwise commuting matrices B_{I} in such a way that the normal closure of the field F (g_{11}, g_{12}, ..., g_{nn}) over F coincides with E and so the group G generated by g^{σ}, σ ∈ Γ is an abelian Γstable group of exponent t. Firstly, we determine the eigenvalues that matrices B_{i} should have if g has the prescribed set of eigenvalues. Collecting the given eigenvalues of pairwise commuting semisimple matrices and using the regular representation, we construct a Γ-stable abelian group G for integral parameters given in proposition.

We consider two different cases in our proof.

1) We suppose that F (ζ_{t}) and E are linearly disjoint over F and [E : F] = d. In this case ϕ_{E}(t) = ϕ_{F} (t). Let w_{1} = 1, w_{2}, ..., w_{d} be a basis of E(ζ_{t}) over F (ζ_{t}), and let Γ be the Galois group of E(ζ_{t}) over F (ζ_{t}). Let g be a semisimple d × dmatrix having eigenvalues ζ_{t}, 1, ..., 1. Using the expansion g = B_{1} +w_{2}B_{2} +...+ w_{d}B_{d} we can construct the matrices B_{i}, i = 1, 2, ..., d, and we can prove that the group G generated by g^{γ}, γ ∈ Γ is an abelian Γ-stable group of exponent t. Let us consider the matrix W = [w_{i}^{σ_{j}}]_{i,j} for σ_{1} = 1, σ_{2}, ..., σ_{d} = Γ. Denote by W_{i} the matrix W whose i-th column is replaced by d chosen eigenvalues ζ_{t}, 1, ..., 1 of g. We can calculate

λ_{i} = \frac{detW_{i}}{detW}

and construct matrices B_{i} as regular representations B_{i} = R(λ_{i}) of λ_{i} in E(ζ_{t})/F (ζ_{t}). Let α_{i,j} be the coefficients of the inverse matrix W^{−1} = [α_{i,j}]_{i,j}. Then α_{i1}^{σ_{j}} = α_{ij} and λ_{j} = (ζ_{t} − 1)α_{ij} for i ≠ 1, and λ_{1} = 1 + (ζ_{t} − 1)α_{11}. So λ_{i}^{σ_{j}} = (ζ_{t} − 1)α_{i1}^{σ_{j}} = (ζ_{t} − 1)α_{ij} for i ≠ 1, and λ_{1}^{σ_{j}} = (ζ_{t} − 1)α_{11}^{σ_{j}} + 1 = (ζ_{t} − 1)α_{1j} + 1. Since any linear relation

k_{1}(λ_{1} − 1) + ∑_{i=2}^{d} k_{i}λ_{i} = 0, k_{i} ∈ F (ζ_{t}), i = 1, 2, ..., d

implies the linear relation

k_{1}(λ_{1}^{σ_{j}} − 1) + ∑_{i=2}^{d} k_{i}λ_{i}^{σ_{j}} = 0, k_{i} ∈ F (ζ_{t}), i = 1, 2, ..., d

for all σ_{j} ∈ Γ, this would also imply detW^{−1} = 0, which is impossible. Therefore, λ_{1}−1, λ_{2}, ..., λ_{d} generate the field E(ζ_{t}) overF (ζ_{t}), and so B_{i}−I_{d}, B_{2}, ..., B_{d} generate F (ζ_{t})-span F (ζ_{t})[B_{1}, ..., B_{d}] over F (ζ_{t}). Note that B_{i} can be expressed as a linear combination of g^{σ_{i}}, i = 1, 2, ..., d with coefficients in E: B_{i} = ∑_{j=1}^{d} α_{ij}g^{σ_{j}}. This can be obtained from the system of matrix equations

g^{σ_{j}} = ∑_{i=1}^{d} w_{i}^{σ_{j}}B_{i}, j = 1, 2, ..., d

if we consider B_{i} as indeterminates. Since G has exponent t, F (ζ_{t}) is a splitting field for G, the group generated by all g^{σ}, σ ∈ Γ. Therefore, the dimension of E(ζ_{t})-span E(ζ_{t})G = E(ζ_{t}) ⊗_{F (ζ_{t})} F (ζ_{t})G over E(ζ_{t}) is d, and so F (ζ_{t})- dimension of F (ζ_{t})-span F (ζ_{t})G is also d.

Let us denote by E′ the image of E(ζ_{t}) under the regular representation of E(ζ_{t})/F (ζ_{t}) over F (ζ_{t}). Then A = E(ζ_{t})G = E(ζ_{t}) ⊗_{F (ζ_{t})} F (ζ_{t})G, the E(ζ_{t})-span of G, is the Galois E′-algebra in the sense of [ILF], that is, it is an associative and commutative separable E′-algebra having a normal basis. We can choose idempotents

ε_{i} = \frac{1}{ζ_{t} − 1}(g^{σ_{j}} − I_{d}), j = 1, 2, ..., d

as a normal basis of A over E′ so that ε_{j} = ε_{1}^{σ_{j}}.

We have F (ζ_{t})G = F (ζ_{t})[< g^{σ_{1}}, ..., g^{σ_{d}} >] = F (ζ_{t})[(g − I_{d})^{σ_{1}}, ..., (g − I_{d})^{σ_{d}}], and dim_{F (ζ_{t})}F (ζ_{t})G = d. As the length of the orbit of M = [m_{ij}] = (g − I_{d}) under Γ-action is d, we can use the coefficients of matrices M^{σ_{i}}, i = 1, 2, ..., d to construct an element θ = ∑_{i,j} k_{ij}m_{ij}, k_{ij} ∈ F (ζ_{t}), which generates a normal basis of E(ζ_{t})/F (ζ_{t}). Therefore, for any given α ∈ E(ζ_{t}) we have α = ∑_{i} k_{i}θ^{σ_{i}} for some k_{i} ∈ F (ζ_{t}).

Therefore, our choice of eigenvalues implies that F (ζ_{t})(G) = E(ζ_{t}).

Now, we can apply the regular representation R_{F} of F (ζ_{t}) over F to matrices M = [m_{ij}]_{i,j}, m_{i,j} ∈ F (ζ_{t}) in the following way: R_{F} (M) = [R_{F} (m_{ij})]_{i,j}. So, using R_{F} for all components of matrices B_{i} ∈ M_{n}(F (ζ_{t})) we can obtain an abelian subgroup G ⊂ GL_{m}(E), m = [F (ζ_{t}) : F]d of exponent t which is Γ-stable if we identify the isomorphic Galois groups of the extensions E/F and E(ζ_{t})/F (ζ_{t}). We have again dim_{F} F G = dim_{E}EG, E is again the Galois algebra, and F (G) = E. Now, using the natural embedding of G to GL_{n}(E), n ≥ m, we complete the proof of Theorem in the case 1).

2) We consider the case when the intersection F_{0} = E ∩ F (ζ_{t}) ≠ F. We can use the regular representation R of E over F. Let Γ_{0} = {σ′_{1}, σ′_{2}, ..., σ′_{d}} be the set of some extensions of elements Γ = {σ_{1}, σ_{2}, ..., σ_{d}} to E(ζ_{t})/F, and let w_{1} = 1, w_{2}, ...w_{d} be a basis of E over F. So we can use our previous notation and go through a similar argument as in part 1) of the proof for construction of g = ∑_{i=1}^{d} B_{i}w_{i} and matrices B_{i} as the regular representations R_{0} of eigenvalues

λ_{i} = \frac{detW_{i}}{detW} = ∑_{j=1}^{ϕ_{E}(t)} λ_{ij}ζ^{j}, i = 1, 2, ..., d,

in the following way: we consider

B_{i} = R_{0}(λ_{i}) = ∑_{j=1}^{ϕ_{E}(t)} R(λ_{ij})ζ^{j},

where R is the regular representation of E over F. We also have λ_{1}^{σ′_{j}} = α_{1j} + 1, λ_{i}^{σ′_{j}} = α_{ij} for j = 2, ..., d . Now, if we have any linear relation between the rows of the matrix [α_{ij}(ζ_{t}^{σ′_{j}} −1)]_{i,j}, this would imply a linear relation between its columns, and so the columns of W^{−1} = [α_{ij}] are linearly dependent, and detW^{−1} = 0 which is a contradiction. So, again we obtain that λ_{1} −1, λ_{2}, ..., λ_{d} are linearly independent over F, so dim_{F} F G′ = dim_{F} F [B_{1} − I_{d}, B_{2}, ..., B_{d}] = dim_{E}EG′ = d for G′ generated by g^{σ′_{i}}, i = 1, 2, ..., d. As earlier we can consider the elementwise regular representation R_{E}(B_{i}) of matrices B_{i} in the field extension E(ζ_{t})/E. So we obtain g_{0} = ∑_{i=1}^{d} R_{E}(B_{i})w_{i}, and we can take the group G generated by all g_{0}^{σ_{i}}, i = 1, 2, ..., d. Since [E(ζ_{t}) : F] = [E(ζ_{t}) : E][E : F] = ϕ_{E}(t)d, the dimension n = ϕ_{E}(t)d coincides with the one required in the formulation of Theorem 1. In this way we can construct a Γ-stable group G that satisfies the conditions of Theorem 1.

This completes the proof of Theorem 1.

Proof of Proposition 1. It follows from the proof of Theorem 1.

If T : C → GL_{k}(E) is an irreducible representation of a cyclic group C generated by an element g of order t, its degree is equal to ϕ_{E}(t) if T (g) ≠ I. We have the following decomposition of E-span A = EG:

A = ε_{1}A_{1} + ε_{2}A_{2} + ... + ε_{d}A_{d}

for the primitive idempotents ε_{1}, ..., ε_{d} forming a normal basis of A, and the rank of matrices ε_{i} is ϕ_{E}(t). So the dimension of A over E is ϕ_{E}(t)d. If G is generated by g_{γ}.γ ∈ Γ and its order is minimal, Γ-stability implies that g has d orbits under Γ-action, and so G an abelian group of type (t, ..., t) and order t^{d}.

Proof of Theorem 2. Let H be an abelian Γ-stable subgroup of GL_{t}(E) of exponent t. Let us consider F -span F H and the system of primitive orthogonal idempotents ε_{1}, ε_{2}, …, ε_{r} of F -algebra F H. We can separate the orbits of this idempotents under Γ-action. We obtain a system {ε_{λ}}_{λ∈I} of one-element orbits and a system of orbits having q_{i} elements {ε_{j}, ε_{j}^{σ}, …, ε_{j}^{σ^{q_{i}−1}}}_{j∈J}. We also consider idempotents e_{l} that are sums of all idempotents contained in the l-th orbit and the associated decomposition of

A = F H = e_{1}A_{1} + e_{2}A_{2} + ... + e_{k}A_{k},

so that e_{l} are Γ-stable and correspond to some Γ-stable component of H.

Let G_{0} be the group containing H which is generated by matrices hε_{i} + (I_{t} − ε_{i}), h ∈ H, i = 1, 2, …, r. G_{0} is a finite Γ-stable abelian group of exponent dividing t, the set of eigenvalues for generators of G_{0} is {1, ζ^{α_{i}}} for an appropriate α_{i} ∈ {1, 2, …, t}. The group G_{0} is conjugate in GL_{t}(F) to the direct sum of matrix groups G_{0i}, i ∈ I and G_{0j}, j ∈ J that correspond to the orbits of minimal idempotents e_{l}, as can be seen by decomposing the algebra A = F G_{0} into the sum ⊕_{i=1}^{r}ε_{i}(F G_{0}) of primitive components. Here the components that correspond to the indices i ∈ I are t-roots of 1, and taken together they compose a subgroup of the group of diagonal matrices of an appropriate dimension. The components corresponding to the indices j ∈ J are matrix groups that are irreducible with respect to GL_{q_{i}v}(F)-conjugation, where v denotes the rank of the idempotent of the corresponding component. Let C = e_{l} = (ε + ε^{σ} + ⋯ + ε^{σ^{q_{i}−1}})G_{0} be one of these components. The matrix group C_{1} that corresponds to this component is contained in GL_{q_{i}v}(F) and is generated by its cyclic subgroups corresponding to matrices ε^{σ^{i}}g + (I − ε^{σ^{i}}), g ∈ G_{0}. (Indeed, the component ε^{σ^{i}}G_{0} is generated by the only matrix ε^{σ^{i}}g for some g ∈ G_{0}, otherwise there would exist h_{1}, h_{2} ∈ ε^{σ^{i}}G_{0} which are not powers of each other, and so some matrix h = h_{1}^{α}h_{2}^{β} ≠ ε^{σ^{i}} and one of its eigenvalues is 1; therefore, we could obtain an idempotent ε_{1} = \frac{1}{p} ∑_{i=0}^{p−1} h^{i}, and ε^{σ^{i}} = ε_{1} + (ε^{σ^{i}} − ε_{1}), and this contradicts the primitivity of idempotent ε^{σ^{i}}). Therefore, G_{0} is generated by the matrices h = εg+(I_{q_{i}v}−ε) and all h^{σ^{j}}, j = 1, 2, …, q_{i}−1 for some matrix g from the chosen component C. Let us consider the following expansion of the matrix h in the basis 1, π^{r_{i}}, π^{2r_{i}}, …, π^{(q_{i}−1)r_{i}} of extension E/F:

h = B_{q} + π^{r_{i}}B_{1} + π^{2r_{i}}B_{2} + ⋯ + π^{(q_{i}−1)r_{i}}B_{q_{i}−1}.

We obtain

B_{j} = \frac{1}{q}((π^{−jr_{i}}h) + (π^{−jr_{i}}h)^{σ} + ⋯ + (π^{−jr_{i}}h)^{σ^{q_{i}−1}}), j = 1, 2, …, q_{i} − 1.

Considering eigenvalues of matrices h and B_{j} we obtain: B_{q_{i}} = I_{q_{i}v} + \frac{1}{q}(ζ − 1)I_{q_{i}v}, B_{j} = B_{1}^{j}, and the matrix B_{1} is conjugate in GL_{n}(F) to the matrix

π^{−q_{i}}I_{v} ⊗ e_{q_{i}1} + I_{v} ⊗ e_{12} + I_{v} ⊗ e_{23} + ⋯ + I_{v} ⊗ e_{q_{i}−1,q_{i}},

and it coincides with I_{v} ⊗ B.

So, components e_{i}A determine either a component C_{1} ∈ GL_{vq_{i}}(E) of G_{0} which is just the group G_{v,q_{i}} specified in Theorem 2, or a subgroup of diagonal matrices. This completes the proof of Theorem 2.

Proof of Theorem 3. We can consider the following sums:

S_{(i_{1}...i_{k})} = \frac{1}{q_{1}...q_{k}} ∑_{γ∈Γ}(π_{1}^{−i_{1}}...π_{k}^{−i_{k}}g)^{γ}

and also expansion of g via the basis π_{1}^{i_{1}}π_{2}^{i_{2}}...π_{k}^{i_{k}}, 0 ≤ i_{j} < q_{j}, j = 1, 2, ..., k as

g = ∑_{(i_{1}...i_{k})} Q_{(i_{1}...i_{k})}π_{1}^{i_{1}}...π_{k}^{i_{k}}.

Since π_{i}^{q_{i}} ∈ F and B_{(i_{1}...i_{k})} ⊂ GL_{n}(F) we obtain S_{(i_{i},...,i_{k})} = Q_{(i_{1},...,i_{k})} = \frac{1}{q}B_{(i_{1},...,i_{k})}(ζ − 1) + δI_{n} for δ = 0 if (i_{1}, ..., i_{k}) ≠ 0 and δ = 0 if (i_{1}, ..., i_{k}) = 0.

Let G be an arbitrary subgroup G ⊂ GL_{n}(F) given in Theorem 2. We denote by E_{0} = F (G) the field obtained by adjoining to F all matrix coefficients of all elements g ∈ G. Let the Galois group Γ of E_{0}/F be the direct sum of cyclic subgroups of orders q_{1}, ..., q_{k} and let Γ_{1}, ...,Γ_{q} be generated by automorphisms σ_{1}, ..., σ_{q} respectively. We consider the system of primitive orthogonal idempotents ε_{1}, ..., ε_{r} of the F -algebra F G and the Γ-orbits of each ε_{i}. So F G splits into a direct sum of irreducible components, and G is conjugate in GL_{n}(F) to a direct sum of GL_{n}(F)-irreducible components. So we can restrict ourselves by considering the particular case of GL_{n}(F)- irreducible group G. Let us represent g as

g = \frac{1}{q} ∑_{(i_{1}...i_{k})}((ζ − 1)B_{(i_{1}...i_{k})}π_{1}^{i_{1}}...π_{k}^{i_{k}} + I_{n}),

and let us introduce the matrices C_{i} given as coefficients in the expansion g = ∑_{j=0}^{q−1} C_{j}π_{k}^{j}. On the other part, matrices C_{i} can be again expanded as C_{i} = ∑_{j=0}^{q_{k−1}−1} C_{ij}π_{k−1}^{j} and so on. Finally, we can obtain the coefficients C_{(j_{1},...,j_{k})} = Q_{(j_{1},...,j_{k})}. Let us denote by I, J the indices (i_{1}, ..., i_{l}), (j_{1}, ..., j_{l}). Let F_{l} be the subfield of elements in E_{0} fixed by σ_{l}, ..., σ_{k}. Then C_{i} can be given by the formula C_{I} = ∑_{i_{l}=0}^{q_{l}−1}(π^{−i_{l}}C_{J})^{σ_{l}^{i_{l}}}. Using conjugation by matrices from GL_{n}(F_{i}) we can transform C_{I} to the direct sum of matrices C′_{I} = ∑_{j_{l}=0}^{q_{l}−1} π_{l}^{j_{i}}M_{I} for

M_{I} = | \begin{matrix}0_{p} & I_{p} & 0_{p} … & 0_{p} \\ 0_{p} & 0_{p} & I_{p} … & 0_{p} \\ . & . & . \\ & & & I_{p} \\ π_{l}^{−q_{l}}C_{J}^{q_{l}} & 0_{p} & 0_{p} … & 0_{p} \end{matrix} |

I = (i_{1}, ..., i_{l}) for i_{l} ≠ 0 or to M_{l} = diag(C_{J}, ..., C_{J}) if i_{l} = 0.

We can repeat this construction for all indices J and I and l ≤ k, (l ≥ 1) and descend the chain of fields F_{l} ⊃ F_{l−1} ⊃ ... ⊃ F_{1} for the coefficients C_{(i_{1},...,i_{k})} in the expansion of g in the basis of E/F. This allows us to complete the proof if we can transform all other matrices h ∈ G to the same form simultaneously. We can do it using the following lemma (for the general theory of commuting matrices (see [G], chapter 8 or [ST]).

Lemma. Let the set of matrices M be some block-diagonal matrices diag(M_{1}, ..., M_{k}) which are contained in GL_{s}(K) for a field K, and let the matrices M_{i} ∈ GL_{s_{i}}(K) form a GL_{s}(K)-irreducible component of degree s_{i}. Then all matrices h ∈ GL_{s}(K) that commute elementwise with matrices in M, can be written as [g_{ij}] for s_{i} × s_{i}-matrices g_{ij} that are nondegenerate if components M_{i} and M_{j} are equivalent and a zero matrix otherwise.

The proof of this lemma follows immediately from the Schur lemma. According to the Schur lemma, any linear operator ρ : K^{n} → K^{m} that satisfies the condition ρδ_{1}(g) = δ_{2}(g)ρ for two irreducible representations g → δ_{1}(g) ∈ GL_{n}(K) and g → δ_{2}(g) ∈ GL_{m}(K) over K and all g, is a zero operator if δ_{1} and δ_{2} are not equivalent and a nondegenerate operator otherwise.

Note. Matrices B_{(i_{1},...,i_{r})} in Theorem 3 can be conjugateded to a different form. Let us consider an l_{i} × l_{i}-matrix

M_{i} = ∑_{i=1}^{q_{i}−1} I_{m} ⊗ e_{i,i+1} + I_{m} ⊗ e_{q_{i}1}

for l_{i} = mq_{1}q_{2}...q_{i} and a matrix B′_{j} = B_{j}. Let

B′_{(j_{i},...,j_{i})} = B′_{(j_{1},...,j_{i−1})} ⊗ (M_{i})^{j_{i}}.

It is easy to see that B′_{(j_{1},...,j_{i})} is conjugate in GL_{l_{i}}(F) to the matrix B_{(j_{1},...,j_{i})} which is given in Theorem 3. Indeed, if

T = ∑_{i=1}^{q_{i}−1}(I_{m} ⊗ A) ⊗ e_{i,i+1} + (I_{m} ⊗ A) ⊗ e_{n1}

and D = diag(I_{m}, I_{m} ⊗ A, ..., I_{m} ⊗ A^{q_{i}−1}) then

DT D^{−1} = ∑_{i=1}^{q_{i}−1} I_{m} ⊗ e_{i,i+1} + (I_{m} ⊗ A^{q_{i}}) ⊗ e_{n1}

for a nondegenerate m × m matrix A. Theorem 3 remains valid if we replace matrices B_{(j_{1},...,j_{k})} with the matrices B′_{(j_{1},...,j_{k})} and use a similar argument as in the proof of Theorem 3.

The result obtained in Theorem 3 can be generalized for solvable extensions E/F with Galois groups Γ if we generalize the construction of matrices B_{i} in Theorem 3. For this we can suppose that F contains primitive roots of 1 of degrees t and q = [E : F]. We can take subnormal series Γ = Γ_{0} ⊃ Γ_{1} ⊃ ... ⊃ Γ_{v} = 1 having cyclic factors Γ_{i}/Γ_{i+1} and the corresponding chain of fields F = K_{0} ⊂ K_{1} ⊂ ... ⊂ K_{v} = E. We can choose an appropriate basis π_{j}^{k}, k = 0, 1, ..., q_{j} − 1 for the field extension K_{j+1}/K_{j} such that π_{j}^{q_{j}} ∈ K_{j} and appropriate generators of Galois groups act on π_{j} by ζ_{j}-multiplication. Using the regular representation of E in F -basis π_{1}^{α_{1}}π_{2}^{α_{2}}...π_{v}^{α_{v}}, 0 ≤ α_{i} < q_{i} of E we can construct the matrices-coefficients R_{(i_{1},...,i_{v})} in the expansion of g:

g = ∑_{(i_{1},...,i_{v})} R_{(i_{1},...,i_{v})}π_{1}^{α_{1}}π_{2}^{α_{2}}...π_{v}^{α_{v}}

in such a way that g^{t} = I and gg^{σ} = g^{σ}g for all σ ∈ Γ. So matrices I_{m} ⊗ g^{σ}, σ ∈ Γ generate a finite abelian Γ-stable matrix group of exponent t. The constructions from Theorems 1 and 2 give an explicite form of R_{(i_{1},...,i_{v})}, and the results analogous to the Theorems 2 and 3 can be obtained for arbitrary Γ-stable abelian matrix groups of exponent t.

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Belarusian State Pedagogical University, Minsk, Belarus Sovetskaya str. 18, 220050 Minsk, Belarus e-mail: malinin@bspu.unibel.by