Hans Zassenhaus

On Hensel factorization, I

Abstract A p-adic method for the constructive factorization of monic polynomials over a dedekind ring o and the ideal theory of o [x] are developed.

On the identification of a Lie algebra given by its structure constants. I. Direct decompositions, levi decompositions, and nilradicals

Abstract Methods are given for identifying a Lie algebra L , given by its structure constants. The identification involves a transformation to a “canonical” basis, in which the structure becomes obvious. Thus, decomposable Lie algebras are already decomposed into direct sums of indecomposable ones. An indecomposable Lie algebra that is not simple or solvable has its radical exhibited. A solvable algebra has its nilradical displayed. The methods all lead to simple algorithms that have been implemented as computer programs, involving some symbolic manipulations.

Solvable Lie algebras of dimension ⩽4 over perfect fields

Abstract The solvable Lie algebras of dimension not greater than four over a perfect field of reference are described in terms of their nilpotent frame, counted over finite fields of q elements, and classified over the real and complex number fields. The number dq,n of nonisomorphic solvable Lie algebras of dimension n over F q roughly speaking grows as a polynomial in n with qn−2 as highest term. But there are smaller additional terms depending on the residue class of q modulo (n−1)! if n⩽4.

Group Rings Whose Units Form an FC-Group

In this paper we give necessary and sufficient conditions for the unit group OIl7LG, of the integral group ring TLGof a group G, to be an FC-group. A group is called an FC-group if all its conjugacy classes are finite. Our second result proves that if K is a field of characteristic 0 then OIl KG is an FC-group if and only if T(G), the set of torsion elements of G, is a finite central subgroup of G. In [7] the groups G with OIl7LGnilpotent are characterised and related results mentioned. We state the present results. (1.1) Theorem. OIl7LGis an FC-group if and only if G torsion subgroup T satisfies one of the following: (1.2) T is central in G, (1.3) Tis abelian non-central andfor xEG is an F C-group and its

Three proofs of Minkowski's second inequality in the geometry of numbers

Let K be a bounded, open convex set in euclidean n-space R n , symmetric in the origin 0. Further let L be a lattice in R n containing 0 and put m 4 =infimum u i i=1,2,.....,n; extended over all positive real numbers u i for which u i K contains i linearly independent points of L. Denote the Jordan content of K by V(K) and the determinant of L by d(L). Minkowskis second inequality in the geometry of numbers states that m 1 m 2 ...m n V(K)≦ 2 n d(L) Minkowskis original proof has been simplified by Weyl [6] and Cassels [7] and a different proof hasbeen given by Davenport [1].

Maximal abelian subalgebras of complex orthogonal lie algebras

Abstract The study of maximal abelian subalgebras (MASAs) of o( n ,C) is reduced to the study of orthogonally indecomposable MASAs. Their classification is in turn reduced to that of maximal abelian nilpotent subalgebras (MANSs) of sl( n /2,C) (for n even) and of o( n ,C) itself. General results and guidelines are given, making it possible to construct all MANSs, and hence all MASAs, for any given n . Representatives of all O( n ,C) conjugacy classes of MASAs of o( n ,C) are constructed for 2⩽ n ⩽7, as an application of the general techniques.

Über verschränkte Produktordnungen

Abstract This paper, which is dedicated to Emmy Noether on the occasion of the centenary of her birthday, is concerned with the arithmetics of crossed products. In particular, the definition of a crossed product order is tightened and it is shown that such orders are “one-headed,” i.e., that the “idealizer of radical” method of embedding always leads to the same hereditary order.

Ein Algorithmus zur Berechnung einer Minimalbasis über Gegebener Ordnung

Fur viele algebraische und zahlentheoretische Anwendungen ist es erforderlich, eine gegebene Ordnung in eine Maximalordnung einzubetten. Obwohl es bekannt ist, das sich das konstruktiv erreichen last, wird, wie ich denke, ein wohldefinierter Algorithmus, der weniger speziell als der Berwick’sehe ist und im allgemeinen schneller zum Ziele fuhrt, von Interesse sein.

The construction of solvable lie algebras from equidimensional nilpotent algebras

Abstract The study of gradings of solvable Lie algebras L of finite dimensionover a field F of zero characteristic led the authors to the discovery of equidimensional nilpotent algebras L ∗ uniquely determined by L up to F -isomorphy. Conversely, for any finite dimensional Lie algebra L ∗ over F an algorithm is developed which yields in parametric form all solvable Lie algebras L determining L ∗ as the corresponding nilpotent algebra. The exposition is independent of Lie grading theory. It organizes in a novel way the classification of solvable Lie algebras of given dimension around the same task for nilpotent algebras. Every isomorphy class of solvable F -algebras is obtained in this way.

Some empirical observations on primitive roots

Abstract Two conjectures about primitive roots are given. These conjectures are supported by empirical evidence obtained by using a digital computer.