A valuation criterion for normal bases in elementary abelian extensions

Let p be a prime number, and let K be a finite extension ofthe field _p of p-adic numbers. Let N be a fully ramified, elementaryabelian extension of K. Under a mild hypothesis on the extensionN/K, we show that every element of N with valuation congruentmod [N:K] to the largest lower ramification number of N/K generatesa normal basis for N over K.     

On some computational problems in finite abelian groups

We present new algorithms for computing orders of elements, discrete logarithms, and structures of finite abelian groups. We estimate the computational complexity and storage requirements, and we explicitly determine the -constants and -constants. We implemented the algorithms for class groups of imaginary quadratic orders and present a selection of our experimental results. Our algorithms are based on a modification of Shanks' baby-step giant-step strategy, and have the advantage that their computational complexity and storage requirements are relative to the actual order, discrete logarithm, or size of the group, rather than relative to an upper bound on the group order.

     

Class numbers of some abelian extensions of rational function fields

Let be a monic irreducible polynomial. In this paper we generalize the determinant formula for of Bae and Kang and the formula for of Jung and Ahn to any subfields of the cyclotomic function field By using these formulas, we calculate the class numbers of all subfields of when and are small.

     

Explicit norm one elements for ring actions of finite abelian groups

It is known that the norm map N _G for the action of a finite groupG on a ringR is surjective if and only if for every elementary abelian subgroupU ofG the norm map N _U is surjective. Equivalently, there exists an elementx _G ∈R satisfying N _G (x _G )=1 if and only if for every elementary abelian subgroupU there exists an elementx _U ∈R such that N _U (x _U )=1. When the ringR is noncommutative, it is an open problem to find an explicit formula forx _G in terms of the elementsx _U . We solve this problem when the groupG is abelian. The main part of the proof, which was inspired by cohomological considerations, deals with the case whenG is a cyclicp-group. Supported by TMR-Grant ERB FMRX-CT97-0100 of the European Union.     

Nonabelian 1-cohomologies and conjugacy of finite subgroups in some extensions

We describe the conjugacy classes of finite subgroups in some split extensions using the notion of 1-cocycle and 1-coboundary with values in a noncommutative group. We prove that each finite subgroup in the automorphism group of a free Lie algebra of rank 3 is conjugated with a subgroup of the linear automorphism group provided that the group order does not divide the characteristic of the ground field.     

The expected number of random elements to generate a finite abelian group

Suppose G is a finite abelian group with minimal number of generators r. It is shown that the expected number of elements from G (chosen independently and with the uniform distribution) so that the elements chosen generate G is less than r + where= 2118456563...The constant is explicitly described in terms of the Riemann zeta-function and is best possible.