Polynomials with Dp as Galois group

A useful criterion characterizing a monic irreducible polynomial over $\text{Q}$ with Galois group D_p (the dihedral group of order 2p, p: prime) is given by making use of the geometry of D_p, i.e., D_p is the symmetry group of the regular p-gon. We derive explicit numerical examples of polynomials with dihedral Galois groups D₅ and D₇.     

D1AD2 theorems for multidimensional matrices

A new technique for proving D₁AD₂ theorems is given. Also a generalization of D₁AD₂ theorems to multidimensional matrices is indicated.     

A parametric family of quintic Thue equations

For an integral parameter we investigate the family of Thue equations

originating from Emma Lehmer's family of quintic fields, and show that for the only solutions are the trivial ones with or . Our arguments contain some new ideas in comparison with the standard methods for Thue families, which gives this family a special interest.

     

On the average depth of a prefix of the Dycklanguage D1

If D₁ is the Dycklanguage with one type of bracket then the level of a bracket in a word w ∈ D₁ is defined as the number of preceding opening brackets minus the number of preceding closing brackets. The depth of a Dyckword w is the maximum level of a bracket appearing in w. In this paper we derive an explicit expression for the average depth of a prefix of length n of the Dycklanguage D₁ and show that the average depth is given for all ε>0 by

$\text{(ln2)}\text{2πn}\text{?}\text{3}\text{2}\text{+}\text{1}\text{8}\text{2π}\text{n}\text{[4ln2?(?1)}{}^{\mathrm{n}}\text{(2ln2?1)]+O(}\text{ln n}\text{n}{}^{\mathrm{1.5?\; ?}}$

. The variance is asymptotically for all ?>0

$\text{(}\text{π}\text{3}\text{?2 ln}{}^2\text{2)πn+}\text{5π}{}^2\text{18}\text{?2πln}{}^2\text{2+}\text{1}\text{12}\text{?π(?1)}{}^{\mathrm{n}}\text{(}\text{π}\text{24}\text{? ln}{}^2\text{2+}\text{1}\text{2}\text{ln2)+O(}\text{ln n}\text{n}{}^{\mathrm{1??}}$

. Furthermore, we derive several enumeration results describing the distribution of the number of certain prefixes of length n.     

Rationality problem of GL4 group actions

Let K be any field which may not be algebraically closed, V be a four-dimensional vector space over K, σ∈GL(V) where the order of σ may be finite or infinite, f(T)∈K[T] be the characteristic polynomial of σ. Let α, αβ₁, αβ₂, αβ₃ be the four roots of f(T)=0 in some extension field of K.Theorem 1.BothK(V)^〈σ〉andare rational (=purelytranscendental) overKif at least one of the following conditions is satisfied: (i) charK=2, (ii) f(T) is a reducible or inseparable polynomial inK[T], (iii) not all ofβ₁,β₂,β₃are roots of unity, (iv) iff(T) is separable irreducible, then the Galois group off(T) overKis not isomorphic to the dihedral group of order 8 or the Klein four group.Theorem 2.Suppose that allβ_iare roots of unity andf(T)∈K[T] is separable irreducible. (a) If the Galois group off(T) is isomorphic to the dihedral group of order 8, then bothK(V)^〈σ〉andare not stably rational overK. (b) When the Galois group off(T) is isomorphic to the Klein four group, then a necessary and sufficient condition for rationality ofK(V)^〈σ〉andis provided. (See Theorem 1.5. for details.)     

K2 and K3 of the circle

Let k be $\text{Z}\text{[}\text{1}\text{2}\text{]}$, $\text{Q}$ or $\text{R}$, and set $\text{A =}\text{k[x,y]}\text{(x}{}^2\text{+ y}{}^2\text{? 1)}$. We compute K₂(A) and K₃(A). Our method is to construct a map $\text{? : K}{}_1\text{(k[i])→K}{}_{\mathrm{1\; +\; 1}}\text{(A)}$ and compare this to a localization sequence.We give three applications. We show that ? accounts for the primitive elements in K₂(A), and compare our results to computations of Bloch [1] for group schemes. Secondly, we consider the problem of basepoint independence, and indicate the interplay of geometry upon the K-theory of affine schemes obtained by glueing points of Spec(A). Third, we can iterate the construction to compute the K-theory of the torus ring A ?_kA.     

A note on Banach spaces containing c0 or C∞

It is shown that every separable Banach space X containing a subspace isomorphic to c₀ has a subspace Y with basis such that $\text{X}\text{Y}\text{~}\text{c}{}_0\text{C}{}_{\mathrm{\infty }}$ and the latter space has a shrinking basis and an unconditional FDD. Moreover, it is shown that X ⊕ C_∞ has a basis if X has the bounded approximation property.     

K2, K3 and nilpotent ideals

In this note is given an explicit computation method to decide about the minimality of a word in the free group of rank two. A cohomological use of this method is to yield words defining one relator duality groups.     

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.     

Groups with T1 primitive ideal spaces

We give a simple necessary and sufficient condition for the group C¹-algebra of a connected locally compact group to have a T₁ primitive ideal space, i.e., to have the property that all primitive ideals are maximal. A companion result settles the same question almost entirely for almost connected groups. As a by-product of the method used, we show that every point in the primitive ideal space of the group C¹-algebra of a connected locally compact group is at least locally closed. Finally, we obtain an analog of these results for discrete finitely generated groups; in particular the primitive ideal space of the group C¹-algebra of a discrete finitely generated solvable group is T₁ if and only if the group is a finite extension of a nilpotent group.