On the 2-class field tower of some imaginary biquadratic number fields

Let be an imaginary biquadratic number field with Cl_k,2, the 2-class group of k, isomorphic to Z/2Z × Z/2^mZ, m > 1, with q a prime congruent to 3 mod 4 and d a square-free positive integer relatively prime to q. For a number of fields k of the above type we determine if the 2-class field tower of k has length greater than or equal to 2. To establish these results we utilize capitulation of ideal classes in the three unramified quadratic extensions of k, ambiguous class number formulas, results concerning the fundamental units of real biquadratic number fields, and criteria for imaginary quadratic number fields to have 2-class field tower length 1. 2000 Mathematics Subject Classification Primary—11R29     

Zur Berechnung der Kapitulation in bikubischen Zahlkörpern

The capitulation kernel is the kernel of the natural extension homomorphism of the ideal class groups in a extension K|k of number fields. In this paper K is a non-cyclic Galois field of degree 6 over the rationals and k is its quadratic subfield. Two different methods of computing the capitulation kernel are discussed. Both depend on the relationship between capitulation and unit structure. The paper closes with two tables. They contain the capitulation kernel for all ramified extensions K|k having cubic discriminants between –20000 and 100000.     

Partial Maps with Domain and Range: Extending Schein's Representation

Let A be a *-algebra. An additive mapping E : A → A is called a Jordan *-derivation if E(X²) = E(x)x^*+xE(x) holds, for all x 6 A. These mappings have been extensively studied in the last 6 years by Bresar, Semrl, Vukman and Zalar because they are closely connected with the problem of representability of quadratic functionals by sesquilinear forms. This study was, however, always in the setting of associative rings. In the present paper we study Jordan *-derivations on the Cayley-Dickson algebra of octonions, which is not associative. Our first main result is that every Jordan *-derivation on the octonion algebra is of the form E(x)=ax*-xa. In the terminology of earlier papers this means that every Jordan *-derivation on the octonion algebra is inner. This generalizes the known fact that Jordan *-derivations on complex and quaternion algebras are inner. Our second main result is a representation theorem for quadratic functionals on octonion modules. Its proof uses the result mentioned above on Jordan *-derivations.     

Hilbert integrals on Siegel domains of type II

The Šilov boundary of a Siegel domain of type II is equivalent to a 2-step nilpotent Lie group. In this paper, we mainly study the L^p-boundness of the Hilbert integrals on Siegel domains of type II by using F. Ricci and E.M. Stein's result about singular integrals on Lie groups^[1]. This is the generalization of part of the work done by P.H.Phong and E.M.Stein in [2]     

Normal integral bases in quadratic and cyclic cubic extensions of quadratic fields

Let K be a number field and let G be a finite abelian group. We call K a Hilbert-Speiser field of type G if, and only if, every tamely ramified normal extension L/K with Galois group isomorphic to G has a normal integral basis. Now let C₂ and C₃ denote the cyclic groups of order 2 and 3, respectively. Firstly, we show that among all imaginary quadratic fields, there are exactly three Hilbert-Speiser fields of type $C_{2}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-1, -3, -7\}$. Secondly, we give some necessary and sufficient conditions for a real quadratic field $K = \mathbb{Q}(\sqrt {m})$ to be a Hilbert-Speiser field of type C₂. These conditions are in terms of the congruence class of m modulo 4 or 8, the fundamental unit of K, and the class number of K. Finally, we show that among all quadratic number fields, there are exactly eight Hilbert-Speiser fields of type $C_{3}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-11,-3, -2, 2, 5, 17, 41, 89\}$.Received: 2 April 2002     

On the equivalence of quadratic APN functions

Establishing the CCZ-equivalence of a pair of APN functions is generally quite difficult. In some cases, when seeking to show that a putative new infinite family of APN functions is CCZ inequivalent to an already known family, we rely on computer calculation for small values of n. In this paper we present a method to prove the inequivalence of quadratic APN functions with the Gold functions. Our main result is that a quadratic function is CCZ-equivalent to the APN Gold function x^2r+1{x^{2^r+1}} if and only if it is EA-equivalent to that Gold function. As an application of this result, we prove that a trinomial family of APN functions that exist on finite fields of order 2 ⁿ where n ≡ 2 mod 4 are CCZ inequivalent to the Gold functions. The proof relies on some knowledge of the automorphism group of a code associated with such a function.     

Imaginary Quadratic Fields with Ono Number 3

In this article, we prove that an imaginary quadratic field F has the ideal class group isomorphic to ?/2? ⊕ ?/2? if and only if the Ono number of F is 3 and F has exactly 3 ramified primes under the Extended Riemann Hypothesis (ERH). In addition, we give the list of all imaginary quadratic fields with Ono number 3.     

Niveau hermitien de Certaines algèbres de quaternions

Let D [d] =(a,b/F) a quaternion divisior algebra over a field F of characteristic ? 2. Denote 1, i, j , k the basis of D, such that i²[d] n, j²[d] b, ij [d] -ji [d] k and A :D → D the involution given by i [d] -i, j [d] j (and k [d] k). In [LE] D. LEWIS asks the following question :Does there exist a quadratic Pfister form [S p. 721 [d] such that the hermitian form [d] [d] D is isotropic over (D, [d]) but not hyperbolic &; In this note, we show that the answer of this question is negative, so that the hermitien level [§I], when it is finite, of (D, A) is a power of two. This result holds for quaternion algebras with standard involution [LE].     

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.     

Quadratic spline-on-spline

Recently Dolezal and Tewarson [2], and Papamichael and Soares [3] have considered the cubic spline-on-spline for the purpose of the approximating the derivatives y⁽²⁾,y⁽³⁾, and y⁽⁴⁾. In this paper their ideas have been extended and the quadratic spline-on-spline has been established for the same purpose. This technique yields better results than the traditional process using a single quadratic spline.     

Intertwining operators for L 2(E)

Let (E,Q) be a finite dimensional quadratic vector space over a finite field. For the natural representation -π of the isometry group G of (E,Q) in the space L ²(E) of all complex valued functions on E, we analyse when the intertwining algebra of π is generated by just one averaging operator.     

A mathematical model for the 3d location estimation of 2D echocardiography data

The recovery of 3D left ventricle(LV) shape using 2D echocardiography is very attractable topic in the field of ultrasound imaging. In this paper, we propose a mathematical model to determine the 3D position of LV contours extracted from multiple 2D echocardiography images. We formulate the proposed model as a non-convex constrained minimization problem. To solve it, we propose a proximal alternating minimization algorithm with a solver OPTI for quadratically constrained quadratic program. For validating the proposed model, numerical experiments are performed with real ultrasound data. The experimental results show that the proposed model is promising and available for real echocardiography data.     

Small zeros of quadratic forms with linear conditions

Given a quadratic form and M linear forms in N+1 variables with coefficients in a number field K, suppose that there exists a point in K^N+1 at which the quadratic form vanishes and all the linear forms do not. Then we show that there exists a point like this of relatively small height. This generalizes a result of D.W. Masser.     

Trace formulae and applications to class numbers

In this paper we compute the trace formula for Hecke operators acting on automorphic forms on the hyperbolic 3-space for the group PSL₂( $\mathcal{O}_K $ ) with $\mathcal{O}_K $ being the ring of integers of an imaginary quadratic number field K of class number H _K > 1. Furthermore, as a corollary we obtain an asymptotic result for class numbers of binary quadratic forms.     

Certain systems of bilinear forms whose minimal algorithms are all quadratic

In this paper we prove that every minimal algorithm for computing a bilinear form is quadratic. It is then easy to show that for certain systems of bilinear forms all minimal algorithms are quadratic. One such system is for computing the product of two arbitrary elements in a finite algebraic extension field. This result, together with the results of the author in (E. Feig, to appear) and those of Winograd (Theoret. Comput. Sci.8 (1979), 359–377), completely characterize all minimal algorithms for computing products in these fields.     

Quadratic Differences that Depend on the Product of Arguments

In this paper, we determine all functions ?, defined on a field K (belonging to a certain class) and taking values in an abelian group, such that the quadratic difference ?(x + y) + ?(x ? y) ? 2?(x) ? 2?(y) depends only on the product xy for all x, y ∈ K. Using this result, we find the general solution of the functional equation ?₁(x + y) + ?₂(x ? y) = ?₃(x) + ?₄(y) + g(xy).     

Class groups of quadratic fields of 3-rank at least 2: Effective bounds

In this paper, we give parametric families of both real and complex quadratic number fields whose class group has 3-rank at least 2. As a consequence, we obtain that for all large positive real numbers x, the number of both real and complex quadratic fields whose class group has 3-rank at least 2 and absolute value of the discriminant ?x is >cx1/3, where c is some positive constant.