A parametric family of quintic polynomials with Galois group D5

We give a parametric family of quintic polynomials of the form x⁵ + ax + b (a, b ∈ $\text{Q}$) with dihedral Galois group D₅. Some properties of the fields defined by these polynomials are also described.     

A note on l-class groups of certain algebraic number fields

The structure of ideal class groups of number fields is investigated in the following three cases: (i) Abelian extensions of number fields whose Galois groups are of type (p, p); (ii) non-Galois extensions $\text{Q(p}\text{d}{}_0\text{3}\text{,p}\text{d}{}_1\text{3}\text{)}$ of degree p² over Q; (iii) dihedral extensions of degree 2^{n + 1} over Q. It is shown that it is possible to obtain class number relations by group-theoretic methods. Subgroups of ideal class groups whose orders are prime to the extension degree are considered.     

Sur la structure galoisienne du groupe des unités d'un corps abélien réel de type (p, p)

Let p be a rational prime. We classify those $\text{Z}$[($\text{Z}$/p$\text{Z}$)²]-modules arising as submodules of the units (mod. torsion) of a real abelian field K with Galois group ($\text{Z}$/p$\text{Z}$)², up to isomorphism and up to genus. Explicit results are given when p is 2 or 3. We apply our classification to discuss the existence of a Minkowski unit in K for arbitrary p.     

Equations with absolute Galois group isomorphic to An

Criteria are given for polynomials of the type Xⁿ + aX³ + bX² + cX + d, to have Galois group over any finite number field isomorphic to A_n. We use them to construct, for every n, infinitely many polynomials with absolute Galois group isomorphic to A_n, covering so, the case n even, $\text{4 ? n}$, for which explicit equations were not known.     

Rickart C1 algebras,II

The theory of inner-outer factorization in the Hardy spaces H^p in the unit disc $\text{D}$ is well known and has many applications. It does not carry over to the spaces H^p on the polydisc $\text{D}$ⁿ or the ball $\text{B}$ⁿ when n > 1. However, for Lumer's Hardy spaces (LH)^p on any simply connected complex analytic manifold, we introduce the notions of internal and external functions and prove that every f? (LH)^p has a factorization f = I_ε × E_ε, where I_ε is internal and E_ε is external, and E_ε? (LH)^p?ε, for any ε > 0. The factorization is not unique and an example of Rudin shows that the ε is needed, at least when $\text{p =}\text{2}\text{m}$, where m is an integer.     

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.     

Linear transformations on matrices: the invariance of generalized permutation matrices—III

Let F be a field, F1 be its multiplicative group, and $\text{H}$ = {H:H is a subgroup of F1 and there do not exist a, b?F1 such that Ha+b?H}. Let D_n be the dihedral group of degree n, H be a nontrivial group in $\text{H}$, and τ_n(H) = {α = (α₁, α₂,…, α_n):α_i?H}. For σ?D_n and α?τ_n(H), let P(σ, α) be the matrix whose (i,j) entry is α_iδ_iσ(j) (i.e., a generalized permutation matrix), and

$\text{P(D}{}_{\mathrm{n}}\text{, H) = {P(σ, α):σ?D}{}_{\mathrm{n}}\text{, α?τ}{}_{\mathrm{n}}\text{(H)}}$

. Let M_n(F) be the vector space of all n×n matrices over F and $\text{T}$P(D_n, H) = {T:T is a linear transformation on M_n (F) to itself and T(P(D_n, H)) = P(D_n, H)}. In this paper we classify all T in $\text{T}$P(D_n, H) and determine the structure of the group $\text{T}$P(D_n, H) (Theorems 1 to 4). An expository version of the main results is given in Sec. 1, and an example is given at the end of the paper.     

A Euler-Poincaré characteristic for the open set lattices of finite topologies

Let $\text{T}$ be a finite topology. If P and Q are open sets of $\text{T}$ (Q may be the null set) then P is a minimal cover of Q provided Q ? P and there does not exist any open set R of $\text{T}$ such that Q ? R ? P. A subcollection $\text{D}$ of the open sets of $\text{T}$ is termed an i-discrete collection of $\text{T}$ provided $\text{D}$ contains every open O ∈ $\text{T}$ with the property that ? $\text{D}$ ? O ? ? $\text{D}$, $\text{D}$ contains exactly i minimal covers of ? $\text{D}$, and provided ?$\text{D}$ = ?{O | O ∈ $\text{D}$ and O is a minimal cover of ? $\text{D}$}. A single open set is a O-discrete collection. The number of distinct i-discrete collections of $\text{T}$ is denoted by p($\text{T}$, i). If there does not exist any i-discrete collection then p($\text{T}$,i) = 0, and this happens trivially for the case when i is greater than the number of points on which $\text{T}$ is defined. The object of this article is to establish the theorem: For any finite topology $\text{T}$, the quantity E($\text{T}$) = Σ_{i = 0}^∞ (?1)ⁱp($\text{T}$, i) = 1.     

Extreme points in sets of positive linear maps on B(H)

Three main results are obtained: (1) If $\text{D}$ is an atomic maximal Abelian subalgebra of $\text{B}$($\text{H}$), $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto $\text{D}$ and h is a complex homomorphism on $\text{D}$, then h ° $\text{P}$ is a pure state on $\text{B}$($\text{H}$). (2) If {P_n} is a sequence of mutually orthogonal projections with rank(P_n) = n and ∑ P_n = I, $\text{P}$ is the projection of $\text{B}$($\text{H}$) onto {P_n}″ given by P(T)=∑trace_n(T)P_n and h is a homomorphism on {P_n}″ such that h(P_n) = 0 for all n then h ° $\text{P}$ induces a type II_∞ factor representation of the Calkin algebra. (3) If $\text{M}$ is a nonatomic maximal Abelian subalgebra of $\text{B}$($\text{H}$) then there is an atomic maximal Abelian subalgebra $\text{D}$ of $\text{B}$($\text{H}$) and a large family {Φ_α} of ¹-homomorphisms from $\text{D}$ onto $\text{M}$ such that for each α, Φ_α ° $\text{P}$ is an extreme point in the set of projections from $\text{B}$($\text{H}$) onto $\text{M}$. (Here $\text{P}$ denotes the projection of $\text{B}$($\text{H}$) onto $\text{D}$.)     

Computing the number of totally positive circular units which are squares

An efficient method for computing the number of invariants of the quotient group $\text{C}\text{C}\text{? T}$, where C is the group of circular units in the maximal real subfield of the qth cyclotomic field, q is a prime, and T is the group of totally positive units, is developed by establishing an isomorphism between $\text{C}\text{C}\text{? T}$ and a specific ideal in the algebra $\text{F[x]}\text{〈x}{}^{\mathrm{p}}\text{+ 1〉}$, where F is the Galois field of two elements and $\text{p =}\text{(q ? 1)}\text{2}$. Based on computations using this method, it is conjectured that every totally positive circular unit is a square if p is a prime.     

Projective unitary positive-energy representations of diff (S1)

Let $\text{D}$ be the group of orientation-preserving diffeomorphisms of the circle S¹. Then $\text{D}$ is Fréchet Lie group with Lie algebra (δ_∞)_R the smooth real vector fields on S¹. Let δ_R be the subalgebra of real vector fields with finite Fourier series. It is proved that every infinitesimally unitary projective positive-energy representation of δ_R integrates to a continuous projective unitary representation of $\text{D}$. This result was conjectured by V. Kac.     

Isomorphisms between Lp-function spaces when p < 1

We prove a number of results concerning isomorphisms between spaces of the type L_p(X), where X is a separable p-Banach space and 0 < p < 1. Our results imply that the quotient of L_p([0, 1] × [0, 1]) by the subspace of functions depending only on the first variable is not isomorphic to L_p, answering a question of N. T. Peck. More generally if $\text{B}$₀ is a sub-σ-algebra of the Borel sets of [0, 1], then $\text{L}{}_{\mathrm{p}}\text{([0, 1])}\text{L}{}_{\mathrm{p}}\text{([0, 1]}$, $\text{B}$₀) is isomorphic to L_p if and only if L_p([0, 1], B₀) is complemented. We also show that L_p has, up to isomorphism, at most one complemented subspace non-isomorphic to L_p and classify completely those spaces X for which $\text{L}{}_{\mathrm{p}}\text{(X) ? L}{}_{\mathrm{p}}$. In particular if $\text{L(L}{}_{\mathrm{p}}\text{, X) = {0}}\text{and}\text{L}{}_{\mathrm{p}}\text{(X) ? L}{}_{\mathrm{p}}$ then $\text{X ? l}{}_{\mathrm{p}}$ or is finite-dimensional. If X has trivial dual and $\text{L}{}_{\mathrm{p}}\text{(X) ? L}{}_{\mathrm{p}}\text{then}\text{X ? L}{}_{\mathrm{p}}$.     

Multipliers from L1(G) to a Lipschitz space

Let G be a metric locally compact Abelian group. We prove that the spaces (L₁, Lip(α, p)), (L₁, lip(α, p)), Lip(α, p) and lip(α, p)^~ are isometrically isomorphic, where Lip(α, p) and lip(α, p) denote the Lipschitz spaces defined on G, (L₁, A) is the space of multipliers from L₁ to A, and lip(α, p)^~ denotes the relative completion of lip(α, p). We also show that $\text{L}{}_1\text{1}\text{Lip}\text{(α, p) =}\text{lip}\text{(α, p) = L}{}_1\text{1}\text{lip}\text{(α, p)}$.     

Transcendence order over Qp in Cp

Let (K, ∥ · ∥) be a valued transcendence degree 1 extension of $\text{Q}$_p. An element x ∈ K transcendental over $\text{Q}$_p is said to have order ≤a (a > 0) if there exists C_x > 0 such that every polynomial P(X) ∈ $\text{Q}$_p [X] satisfies

$\text{?}\text{log;}\text{(P(x))? ?}\text{log}\text{∥P∥+c}{}_{\mathrm{x}}\text{(}\text{deg}\text{P)}{}^{\mathrm{a}}$

when ∥ · ∥ is the Gauss norm on $\text{Q}$_p[X]. No x ∈ $\text{C}$_p can have order ≤α if α < 1 but we construct some x ∈ $\text{C}$_p with order ≤ 1. Furthermore, we prove order ≤α is stable by algebraic extension.     

On the sum ∑k=13 | 2πk − αk |, αk algebraic numbers

If α₁, α₂, α₃ are algebraic numbers satisfying (i) the height of α₁, α₂, α₃ do not exceed H (ii) the degree of the field generated by α₁, α₂, α₃ over the field of rational numbers do not exceed D, then a positive lower bound for

$\text{∑}\text{k=1}\text{3}\text{|2π}{}^{\mathrm{k}}\text{?αk|}$

is determined explicitly (except for an absolute constant) in terms of D and H.     

An integral transform in L2-cohomology for the ladder representations of U(p, q)

Starting from the realization of the Fock space as L²-cohomology of $\text{C}$^{p + q}, $\text{H}$^0,p($\text{C}$^{p + q}) = ⊕_m?$\text{Z}$$\text{H}$_m^0,p($\text{C}$^{p + q}), an integral transform is constructed which is a direct-image mapping from $\text{H}$_m^{0,p($\text{C}$p + q) into the space of holomorphic sections of some vector bundle E_m over M ≈ U(p, q)/(U(q) × U(p)), m ? 0}. The transform intertwines the natural actions of U(p, q) and is injective if m ? 0, so it provides a geometric realization of the ladder representations of U(p, q). The sections in the image of the transform satisfy certain linear differential equations, which are explicitly described. For example, Maxwell's equations are of this form if p = q = 2 and m = 2. Thus, this transform is analogous to the Penrose correspondence.     

Semidirect product division algebras

SupposeD is a division algebra of degreep over its centerF, which contains a primitivep-root of 1. Also supposeD has a maximal separable subfield overF whose Galois group is the semidirect product of the cyclic groupsC _p C _q , whereq=2, 3, 4, or 6 and is relatively prime top (In particular this is the case whenp is prime ≤7 andD has a maximal separable subfield whose Galois group is solvable.) ThenD is cyclic. The proof involves developing a theory of a wider class of algebras, which we call accessible, and proving that they are cyclic.     

A decomposition of R into two homeomorphic rigid parts

Let X be separable, completely metrizable, and dense in itself. We show that if X admits a triple (D₁, D₂, h) of two countable dense subsets D₁ and D₂ and a homeomorphism h: X?D₁ → X?D₂, satisfying some special properties, then there is a rigid subspace A of X such that A is homeomorphic to X?A = h[A]; for $\text{X =}\text{R}$, such atriple is shown to exist.