Self-dual normal bases for infinite galois field extensions

Let L ? K be an infinite Galois field extension with the property that every finite Galois extension M ? K, where L ? M, has a self-dual normal basis. We prove a self-dual normal basis theorem for L ? K when char (K) ≠2.     

Explicit construction of self-dual integral normal bases for the square-root of the inverse different

Let K be a finite extension of Qp, let L/K be a finite abelian Galois extension of odd degree and let OL be the valuation ring of L. We define AL/K to be the unique fractional OL-ideal with square equal to the inverse different of L/K. For p an odd prime and L/Qp contained in certain cyclotomic extensions, Erez has described integral normal bases for AL/Qp that are self-dual with respect to the trace form. Assuming K/Qp to be unramified we generate odd abelian weakly ramified extensions of K using Lubin-Tate formal groups. We then use Dwork's exponential power series to explicitly construct self-dual integral normal bases for the square-root of the inverse different in these extensions.     

Primitive complete normal bases: Existence in certain 2-power extensions and lower bounds

The present paper is a continuation of the author’s work (Hachenberger (2001) [3]) on primitivity and complete normality. For certain 2-power extensions E over a Galois field F_q, we are going to establish the existence of a primitive element which simultaneously generates a normal basis over every intermediate field of E/F_q. The main result is as follows: Letq≡3mod4and letm(q)≥3be the largest integer such that2^m(q)dividesq²−1; ifE=F_q^2l, wherel≥m(q)+3, then there exists a primitive element inEthat is completely normal overF_q.Our method not only shows existence but also gives a fairly large lower bound on the number of primitive completely normal elements. In the above case this number is at least 4⋅(q−1)^2l−2. We are further going to discuss lower bounds on the number of such elements in r-power extensions, where r=2 and q≡1mod4, or where r is an odd prime, or where r is equal to the characteristic of the underlying field.     

Normal integral bases and strict ray class groups modulo 4

Let F be a number field. We construct three tamely ramified quadratic extensions which are ramified at most at some given set of finite primes, such that K₃⊂K₁K₂, both K₁/F and K₂/F have normal integral bases, but K₃/F has no normal integral basis. Since Hilbert-Speiser's theorem yields that every finite and tamely ramified abelian extension over the field of rational numbers has a normal integral basis, it seems that this example is interesting (cf. [5] J. Number Theory 79 (1999) 164; Theorem 2). As we shall explain below, the previous papers (Acta Arith. 106 (2) (2003) 171-181; Abh. Math. Sem. Univ. Hamburg 72 (2002) 217-233) motivated the construction. We prove that if the class number of F is bigger than 1, or the strict ray class group of F modulo 4 has an element of order ?3, then there exist infinitely many triplets (K₁,K₂,K₃) of such fields.     

A method for constructing a self-dual normal basis in odd characteristic extension fields

This paper proposes a useful method for constructing a self-dual normal basis in an arbitrary extension field F_p^m such that 4p does not divide m(p−1) and m is odd. In detail, when the characteristic p and extension degree m satisfies the following conditions (1) and either (2a) or (2b); (1) 2km+1 is a prime number, (2a) the order of p in F_2km+1 is 2km, (2b) 2km and the order of p in F_2km+1 is km, we can consider a class of Gauss period normal bases. Using this Gauss period normal basis, this paper shows a method to construct a self-dual normal basis in the extension field F_p^m.     

A note on integral bases of unramified cyclic extensions of prime degree. II

Let p be a prime number and K a number field containing a primitive p-th root of unity. It is known that an unramified cyclic extension L/K of degree p has a power integral basis if it has a normal integral basis. We show that for all p, the converse is not true in general. Received: 18 July 2000 / Revised version: 18 October 2000     

Different interpretations of the improper integral objective in an infinite horizon control problem

We provide an example of a convex infinite horizon problem with a linear objective functional where the different interpretations of the improper integral in either Lebesgue or Riemann sense lead to different but finite optimal values.     

有限域上存在弱自对偶正规基的一个充要条件

对于将有限域上的自对偶基概念推广到了更一般的弱自对偶的情形,给出了有限域上存在这类正规基的一个充妥条件:设q为素数幂,E=Fqn为q元域F=Fq的n次扩张,N={αi=αqi|i=0,1,…,n-1}为E在F上的一组正规基．则存在c∈F*及r,0≤r≤n-1,使得β=cαr生成N的对偶基的充要条件是以下三者之一成立: (1)q为偶数且n≠0(mod 4);(2) n与q均为奇数;(3)q为奇数,n为偶数,(-1)为F中的非平方元且r为奇数．     

Power integral bases in a parametric family of totally real cyclic quintics

We consider the totally real cyclic quintic fields , generated by a root of the polynomial

Assuming that is square free, we compute explicitly an integral basis and a set of fundamental units of and prove that has a power integral basis only for . For (both values presenting the same field) all generators of power integral bases are computed.

     

Explicit iteration and unbounded solutions for fractional integral boundary value problem on an infinite interval

By employing the monotone iterative technique, we not only establish the existence of the unique solution for a fractional integral boundary value problem on semi-infinite intervals, but also develop an explicit iterative sequence for approximating the solution and give an error estimate for the approximation, which is an important improvement of existing results.     

Abel differential equations admitting a certain first integral

In this work algebro-geometric conditions to have a certain first integral for an Abel differential equation are given. These conditions establish a bridge with classical Galois theory because we transform the differential problem of finding a first integral for an Abel equation into an algebraic problem.     

Normal bases for Hopf-Galois algebras

Let be a Hopf algebra over a commutative ring such that is a finitely generated, projective module over , let be a right -comodule algebra, and let be the subalgebra of -coinvariant elements of . If is a Galois extension of and is a local subalgebra of the center of , then is a cleft right -comodule algebra or, equivalently, there is a normal basis for over .

     

A note on normal bases of ideals in sextic algebraic number field

Let K/Q be a cyclic tamely ramified extension of degree 6, then any ambiguous ideal of K has a normal basis. Supported by grants GACR 201/07/0191, VEGA 2/4138/04, VEGA 1/0084/08.     

Generic extensions and generic polynomials for semidirect products

This paper presents a generalization of a theorem of Saltman on the existence of generic extensions with group AG over an infinite field K, where A is abelian, using less restrictive requirements on A and G. The method is constructive, thereby allowing the explicit construction of generic polynomials for those groups, and it gives new bounds on the generic dimension.Generic polynomials for several small groups are constructed.     

Riesz bases for p-subordinate perturbations of normal operators

For a class of unbounded perturbations of unbounded normal operators, the change of the spectrum is studied and spectral criteria for the existence of a Riesz basis with parentheses of root vectors are established. A Riesz basis without parentheses is obtained under an additional a priori assumption on the spectrum of the perturbed operator. The results are applied to two classes of block operator matrices.     

Nontrivial Galois module structure of cyclotomic fields

We say a tame Galois field extension with Galois group has trivial Galois module structure if the rings of integers have the property that is a free -module. The work of Greither, Replogle, Rubin, and Srivastav shows that for each algebraic number field other than the rational numbers there will exist infinitely many primes so that for each there is a tame Galois field extension of degree so that has nontrivial Galois module structure. However, the proof does not directly yield specific primes for a given algebraic number field For any cyclotomic field we find an explicit so that there is a tame degree extension with nontrivial Galois module structure.

     

Global weak sharp minima for convex (semi-)infinite optimization problems

We mainly consider global weak sharp minima for convex infinite and semi-infinite optimization problems (CIP). In terms of the normal cone, subdifferential and directional derivative, we provide several characterizations for (CIP) to have global weak sharp minimum property.