The construction of SL(2, 3)-polynomials

Octic polynomials over $\text{Z}$ with Galois group SL(2, 3) are constructed. This is done via suited quartic totally real polynomials with group A₄ over $\text{Q}$. A table of the cycle patterns of the imprimitive transitive permutation groups of degree 8 is included.     

On the number of conjugacy classes in SL(2,Z)

Let $\text{H}\text{(t)}$ be the number of conjugacy classes of elements in SL(2, $\text{L}$) with trace t, and let $\text{h}\text{(n)}$ be the number of equivalence classes of binary quadratic forms with discriminant n. Then for t≠±2, $\text{H}\text{(t)=}\text{h}\text{(t}{}^2\text{?4)}$. For all real θ > 0 there is a T(θ) such that whenever |t|>T(θ), $\text{H}\text{(t)>|t|}{}^{\mathrm{1?\theta }}$. There is a c>0 such that for those t such that t²?4 is squarefree, $\text{H}\text{(t)≤c|t|}$.     

A characterization of the closed unital ideals of the Fourier-Stieltjes algebra B(G) of a locally compact amenable group G

Let G be a locally compact amenable group, B(G) its Fourier-Stieltjes algebra and I be a closed ideal of it. In this paper we prove the following result: The ideal I has a unit element iff it is principal. This is the noncommutative version of the Glicksberg-Host-Parreau Theorem. The paper also contains an abstract version of this theorem.     

Invariant means and invariant ideals in L∞(G) for a locally compact group G

For a nondiscrete σ-compact locally compact Hausdorff group G, L_∞(G) is a commutative Banach algebra under pointwise multiplication which has many nonzero proper closed invariant ideals; there is at least a continuum of maximal invariant ideals {N_α} such that N_α1 + N_α2 = L_∞(G) whenever α₁ ≠ α₂. It follows from the construction of these ideals that when G is also amenable as a discrete group, then LIM?TLIM contains at least a continuum of mutually singular elements each of which is singular to any element of TLIM. The supports of left-invariant means are in the maximal ideal space of L_∞(G); the structure of these supports leads to the notion of stationary and transitive maximal ideals. To prove that both these types of maximal ideals are dense among all maximal ideals, one shows that the intersection of all nonzero closed invariant ideals is zero. This is the case even though the intersection of any sequence of closed invariant ideals is not zero and the intersection of all the maximal invariant ideals is not zero.     

Uniqueness of norm on L(G) and C(G) when G is a compact group

Let G be a compact group. If the trivial representation of G is not weakly contained in the left regular representation of G on L₀²(G) and X is either L^p(G) for 1<p?∞ or C(G), then we show that every complete norm |·| on X that makes translations from (X,|·|) into itself continuous is equivalent to ||·||_p or ||·||_∞ respectively. If 1<p?∞ and every left invariant linear functional on L^p(G) is a constant multiple of the Haar integral, then we show that every complete norm |·| on L^p(G) that makes translations from (L^p(G),|·|) into itself continuous and that makes the map t?L_t from G into bounded is equivalent to ||·||_p.     

The equivalence of curves in SL(n, R) and its application to ruled surfaces

The generating system of the differential algebra for invariant differential polynomials with two parametric curves is obtained. Conditions for the equivalence of two parametric curves families are given. We are also proved that the generating differential invariants of two parametric curves are independent. Finally, we reduce the SL(n, R)-equivalent problem for ruled surfaces to that of parametric curves.     

Factorization of Q(h(T)(x)) over a finite field where Q(x) is irreducible and h(T)(x) is linear—I

Let GF(q) be the finite field of order q, let Q(x) be an irreducible polynomial in GF(q)(x), and let h(T)(x) be a linear polynomial in GF(q)[x], where T:x→x^q. We use properties of the linear operator h(T) to give conditions for Q(h(T)(x)) to have a root of arbitrary degree k over GF(q), and we describe how to count the irreducible factors of Q(h(T)(x)) of degree k over GF(q). In addition we compare our results with those Ore which count the number of irreducible factors belonging to a linear polynomial having index k.     

Factorization of Q(h(T)(x)) over a finite field,where Q(x) is irreducible and h(T)(x) is linear II

Let GF(q) be the finite field of order q, let Q(x) be an irreducible polynomial in GF(qⁱ)[x], and let H(x) = h(T)(x) be a linear polynomial in GF(q)[x]. We give the degrees of the irreducible factors of Q(H(x)) in GF(qⁱ)[x], and the number of irreducible factors of each degree. We consider the special cases when H(x) is a trace function, and when h(x) is cyclotomic. Finally, we give several examples.     

Diagonalizing matrices over C(X)

A complete characterization of those compact Hausdorff spaces is given such that for every n, each normal element in the algebra C(X)?M_n of continuous functions from X to $\text{M}$_n can be continuously diagonalized. The conditions are that X be a sub-Stonean space with dim X ? 2 and carries no nontrivial G-bundles over any closed subset, for G a symmetric group or the circle group. In particular, diagonalization is assured on every totally disconnected sub-Stonean space, but also on connected spaces of the form β(Y)/Y, where Y is a simply-connected (noncompact) graph.     

Chords in a circle and linear algebra over GF(2)

A common framework for the two concepts of the title is developed to yield an alternative proof to a theorem of Cohn and Lempel relating the number of orbits of a product of a full cycle by disjoint transpositions to the rank over GF(2) of the associated chord-intersection-matrix.     

Translation planes of order ϱ6 admitting SL(2, ϱ2)

Large numbers of translation planes are constructed which have order ?⁶ and admit a collineation group SL(2, ?²) generated by elations.     

The functional equation P(f)=Q(g) in a p-adic field

Let K be a complete ultrametric algebraically closed field of characteristic π. Let P,Q be in K[x] with P′Q′ not identically 0. Consider two different functions f,g analytic or meromorphic inside a disk |x−a|<r (resp. in all K), satisfying P(f)=Q(g). By applying the Nevanlinna's values distribution Theory in characteristic π, we give sufficient conditions on the zeros of P′,Q′ to assure that both f,g are “bounded” in the disk (resp. are constant). If π≠2 and deg(P)=4, we examine the particular case when Q=λP (λ∈K) and we derive several sets of conditions characterizing the existence of two distinct functions f,g meromorphic in K such that P(f)=λP(g).     

Injective and projective objects in the category of locally compact modules over the ring of integral values of a global field

In the paper injective and projective objects in the category of locally compact modules over the ring of integral values of a global field are described together with the objects of this category possessing injective and projective resolvents. Translated fromMatematicheskie Zametki, Vol. 62, No. 1, pp. 118–123, July, 1997. Translated by A. I. Shtern