On indecomposable algebras of exponent 2

For any n ≥ 3 we give numerous examples of central division algebras of exponent 2 and index 2ⁿ over fields, which do not decompose into a tensor product of two nontrivial central division algebras, and which are sums of n + 1 quaternion algebras in the Brauer group of the field. Also, for any n ≥ 3 and any field k ₀ we construct an extension F/k ₀ and a multiquadratic extension L/F of degree 2ⁿ such that for any proper subextensions L ₁/F and L ₂/F

Potenzieren und Wurzelziehen in rationalen Quaternionenalgebren

LetF be a field not of characteristic 2 andQ =F +F i +F j +F k the quaternion algebra overF whereij = -ji =k andi ² = α andj ² = β with 0 ≠ α, β ∈F fixed. (IfF = ℝ and α = β = - 1 thenQ is the division algebra of the Hamilton quaternions.) IfF = ℚ and Q is a division algebra then by embedding certain quadratic number fields inQ we derive an efficient formula to compute the powers of any quaternion. This formula is even true in general and reads as follows. If a, a₁, a₂, a₃ ∈F andn ∈ ℕ then where ω ig a square root of αa₁ ² + βa ₂ ² - αβa ₃ ² in or overF and andA ₀ =na ^n-1. With the help of this formula and related ones we are able to solve the equationX ⁿ =q for arbitrary quaternionsq and positive integers n in case ofF = ℝ and hence in case ofF ⊂ ℝ as well. IfF = ℝ then the total number of all solutions equals 0, 1, 2, 4,n or ∞. (4 is possible even whenn < 4.) In case ofF = ℚ, which we are primarily interested in, there are always either at most six or infinitely many solutions. Further, for everyq ≠ 0 there is at most one solution provided thatn is odd and not divisible by 3. The questions when there are infinitely many solutions and when there are none can always be decided by checking simple conditions on the radicandq ifF = ℝ. ForF = ℚ the two questions are comprehensively investigatet in a natural connection with ternary and quaternary quadratic rational forms. Finally, by applying some of our theorems on powers and roots of quate-rions we also obtain several nice results in matrix theory. For example, for every k ∈ ℤ the mappingA ↦A ^k on the group of all nonsingular 2-by-2 matrices over ℚ is injective if and only ifk is odd and not divisible by 3.

     

On some elements of the Brauer group of a conic

The main purpose of the paper is to strengthen previous author’s results. Let k be a field of characteristic ≠ 2, n ≥ 2. Suppose that elements are linearly independent over ℤ/2ℤ. We construct a field extension K/k and a quaternion algebra D = (u, v) over K such that

Largest Families Without an <Emphasis Type="Italic">r</Emphasis>-Fork

Let [n] = { 1,2,...,n} be a finite set, a family of its subsets, 2 ≤ r a fixed integer. Suppose that contains no r + 1 distinct members F, G ₁,..., G _r such that F ⊂ G ₁,...,F ⊂ G _r all hold. The maximum size is asymptotically determined up to the second term, improving the result of Tran. The work of the second author was supported by the Hungarian National Foundation for Scientific Research grant numbers NK0621321, AT048826, the Bulgarian National Science Fund under Grant IO-03/2005 and the projects of the European Community: INTAS 04-77-7171, COMBSTRU–HPRN-CT-2002-000278, FIST–MTKD-CT-2004-003006.     

Gelfand-Kirillov dimension under base field extension

LetF ⊂K be a field extension,A be aK-algebra. It is proved that, in general, GK dim _F A≥GK dim _K A+tr _F (K). For commutative algebras or Noetherian P.I. algebras, the equality holds. Two examples are also constructed to show that: (i) there exists an algebraA such that GK dim _F A=GK dim _K A+tr _F (K)+1; (ii) there exists an algebraic extensionF ⊂K and aK-algebraA such that GK dim _F A=∞, but GK dim _K A<∞.     

On the property <Emphasis Type="Italic">D</Emphasis>(2) and a common splitting field of two biquaternion algebras

Let F be a field of characteristic ≠ 2. We say that F possesses the property D(2) if for any quadratic extension L/F and any two binary quadratic forms over F having a common nonzero value over L, this value can be chosen in F. There exist examples of fields of characteristic 0 that do not satisfy the property D(2). However, as far as we know, it is still unknown whether there are such examples of positive characteristic and what is the minimal 2-cohomological dimension of fields for which the property D(2) does not hold. In this note it is shown that if k is a field of characteristic ≠ 2 such that |k*/k*²| ≥ 4, then for the field k(x) the property D(2) does not hold. Using this fact, we construct two biquaternion algebras over a field K = k(x)((t))((u)) such that their sum is a quaternion algebra, but they do not have a common biquadratic (i.e., a field of the kind , where a, b ∈ K*) splitting field. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 338, 2006, pp. 242–250.     

Cohomology of a flag variety as a Bethe algebra

We interpret the equivariant cohomology H_GLn ^*H_{GL_n }^* (ℱ _λ ,ℂ) of a partial flag variety ℱ _λ parametrizing chains of subspaces 0 = F ₀ ⊂ F ₁ ⊂ … ⊂ F _N = ℂ ⁿ , dimF _i /F _i−1 = λ _i , as the Bethe algebra of the -weight subspace of a [t]-module .     

Asymptotic Expansions in Non-central Limit Theorems for Quadratic Forms

We consider quadratic forms of the type

Finiteness conditions in Krull subrings of a ring of polynomials

LetR be a Krull subring of a ring of polynomialsk[x ₁, …, x_n] over a fieldk. We prove that ifR is generated by monomials overk thenr is affine. We also construct an example of a non-affine Krull ringR, such thatk[x, xy]⊂R⊂k[x, y], and a non-Noetherian Krull ringS, such thatk[x, xy, z]⊂S⊂k[x, y, z].     

The generalized Rédei-matrix

Let F be a number field with odd class number, and let E be a quadratic extension of F. Our main aim is to prove that the 4-rank of the class group C(E) of E is equal to m − 1 − rank R _E/F , where m is the number of primes of F ramifying in E, R _E/F is the generalized Rédei-matrix of local Hilbert symbols with coefficients in and the rank is the rank over . We determine the generalized Rédei-matrices R _E/F explicitly for biquadratic number fields E. The research is partly supported by NNSF of China (No. 10371054, No. 10771100) and the Morningside Center of Mathematics in Beijing (MCM).     

Stability preservation theorems

The basic result of the paper is the main theorem worded as follows. Let {ie155-01} be a valued field such that {ie155-02} has characteristic p > 0 and let {ie155-03} be an extension of valued fields satisfying the following conditions: (i) there exists a set {ie155-04} for which {ie155-05} is a separating transcendence basis for a field {ie155-06} over F_R; (ii) Γ _R is p-pure in {ie155-07}, i.e., {ie155-08} does not contain elements of order p; (iii) there exists a set B₁ ⊂ F₀^× such that the family {ie155-09} is linearly independent in the elementary p-group {ie155-10}; (iv) F₀ is algebraic over F(B₀ ⋃ B₁). Then the property of being stable for {ie155-11} implies being stable for {ie155-12}. Supported by the Council for Grants (under RF President) and State Aid of Leading Scientific Schools (grant NSh-344.2008.1) and by RFBR (grant No. 08-01-00442-a). __________ Translated from Algebra i Logika, Vol. 47, No. 3, pp. 269–287, May–June, 2008.     

Implicatively selector sets

Let A⊆N={0,1,2,...} and β be an n-ary Boolean function. We call A a β-implicatively selector (β-IS) set if there exists an n-ary selector general recursive function f such that (∀x₁,...,x_n)(β(χ(x₁),...,χ(x_n))=1⟹f(x₁,...,x_n)∈A), where χ is the characteristic function of A. Let F^(m), m≥1, be the family of all d _m+1 ^* -IS sets, where , F⁽⁰⁾=N, and F^(∞) is the class of all subsets in N. The basic result of the article says that the family of all β-IS sets coincides with one of F^(m), m≥0, or F^(∞), and, moreover, the inclusions F⁽⁰⁾⊂F⁽¹⁾⊂...⊂F^(∞) hold. Translated fromAlgebra i Logika, Vol. 35, No. 2, pp. 145–153, March–April, 1996.     

Geometry of 2 × 2 hermitian matrices

LetD be a division ring which possesses an involution a → α . Assume that is a proper subfield ofD and is contained in the center ofD. It is pointed out that ifD is of characteristic not two, D is either a separable quadratic extension of F or a division ring of generalized quaternions over F and that if D is of characteristic two,D is a separable quadratic extension ofF. Thus the trace map Tr:D → F, a → a + a is always surjective, which is formerly posed as an assumption in the fundamental theorem of n×n hermitian matrices overD when n ≥ 3 and now can be deleted. WhenD is a field, the fundamental theorem of 2 × 2 hermitian matrices overD has already been proved. This paper proves the fundamental theorem of 2×2 hermitian matrices over any division ring of generalized quaternions of characteristic not two This research was completed during a visit to the Academy of Mathematics and System Sciences, Chinese Academy of Sciences.     

Mappings of the sphere to a simply connected space

Fix an m ∈ ℕ, m ≥ 2. Let Y be a simply connected pointed CW-complex, and let B be a finite set of continuous mappings a: S^m → Y respecting the distinguished points. Let Γ(a) ⊂ S^m × Y be the graph of a, and we denote by [a] ∈ π_m(Y) the homotopy class of a. Then for some r ∈ ℕ depending on m only, there exist a finite set E ⊂ S^m × Y and a mapping k: E(r) = {F ⊂ E: |F| ≤ r} → π_m(Y) such that for each a ∈ B we have

Gauss Sum of Index 4： （1） Cyclic Case

Let p be a prime, m ≥ 2, and （m,p（p - 1）） = 1. In this paper, we will calculate explicitly the Gauss sum G（X） = ∑x∈F＊qX（x）ζ^Tp^（x） in the case of [（Z/mZ）＊ ： （p）] = 4, and -1 （不属于） （p）, where q P^f, f =φ（m）/4, X is a multiplicative character of Fq with order m, and T is the trace map for Fq/Fp. Under the assumptions [（Z/mZ）＊ ： （p）] = 4 and 1（不属于） （p）, the decomposition field of p in the cyclotomic field Q（ζm） is an imaginary quartic （abelian） field. And G（X） is an integer in K. We deal with the case where K is cyclic in this oaDer and leave the non-cvclic case to the next paper.     

The 1-Lipschitz mapping between the unit spheres of two Hilbert spaces can be extended to a real linear isometry of the whole space

LetE andF be Hilbert spaces with unit spheresS ₁(E) andS ₁(F). Suppose thatV ₀ S₁(E) →S ₁(F) is a Lipschitz mapping with Lipschitz constantk=1 such that −V ₀[S ₁(E)] ⊂V ₀[S ₁(E)]. Then V₀ can be extended to a real linear isometric mappingV fromE intoF. In particular, every isometric mapping fromS ₁(E) ontoS ₁(F) can be extended to a real linear isometric mapping fromE ontoF.     

Subcontinuity

We give interesting characterizations using subcontinuity. Let X, Y be topological spaces. We study subcontinuity of multifunctions from X to Y and its relations to local compactness, local total boundedness and upper semicontinuity. If Y is regular, then F is subcontinuous iff [`(F)]\bar F is USCO. A uniform space Y is complete iff for every topological space X and for every net {F _a }, F _a ⊂ X × Y, of multifunctions subcontinuous at x ∈ X, uniformly convergent to F, F is subcontinuous at x. A Tychonoff space Y is Čech-complete (resp. G _m-space) iff for every topological space X and every multifunction F ⊂ X × Y the set of points of subcontinuity of F is a G _δ -subset (resp. G _m-subset) of X.     

Towards dimension expanders over finite fields

In this paper we study the problem of explicitly constructing a dimension expander raised by [3]: Let \mathbbFⁿ \mathbb{F}^n be the n dimensional linear space over the field \mathbbF\mathbb{F}. Find a small (ideally constant) set of linear transformations from \mathbbFⁿ \mathbb{F}^n to itself {A _i } _i∈I such that for every linear subspace V ⊂ \mathbbFⁿ \mathbb{F}^n of dimension dim(V)<n/2 we have

Quadratic transformations for a function that generalizes 2F1 and the Askey-Wilson polynomials

In previous papers we introduced and studied a ‘relativistic’ hypergeometric function R(a ₊, a ₋, c; v, ) that satisfies four hyperbolic difference equations of Askey-Wilson type. Specializing the family of couplings c∊ to suitable two-dimensional subfamilies, we obtain doubling identities that may be viewed as generalized quadratic transformations. Specifically, they give rise to a quadratic transformation for ₂ F ₁ in the ‘nonrelativistic’ limit, and they yield quadratic transformations for the Askey-Wilson polynomials when the variables v or are suitably discretized. For the general coupling case, we also study the bearing of several previous results on the Askey-Wilson polynomials. Dedicated to Richard Askey on the occasion of his 70th birthday. 2000 Mathematics Subject Classification Primary—33D45, 39A70     

Une preuve simple des conjectures de Langlands pour GL(n) sur un corps p-adique

Let F be a finite extension of ℚ _p . For each integer n≥1, we construct a bijection from the set ?_F ⁰ (n) of isomorphism classes of irreducible degree n representations of the (absolute) Weil group of F, onto the set ? _F ⁰ (n) of isomorphism classes of smooth irreducible supercuspidal representations of GL _n (F). Those bijections preserve epsilon factors for pairs and hence we obtain a proof of the Langlands conjectures for GL _n over F, which is more direct than Harris and Taylor’s. Our approach is global, and analogous to the derivation of local class field theory from global class field theory. We start with a result of Kottwitz and Clozel on the good reduction of some Shimura varieties and we use a trick of Harris, who constructs non-Galois automorphic induction in certain cases. Oblatum 1-III-1999 & 21-VII-1999 / Published online: 29 November 1999