Klaus Hoechsmann

ALGEBRAS SPLIT BY A GIVEN PURELY INSEPARABLE FIELD

1. Let K be a field of characteristic p O. By a p-algebra we mean a central simple algebra over K whose dimension is a power of p. Although it is known that such an algebra always has a purely inseparable (over K) splitting field E, the role played by E in the structure of the algebra has not been clear. In this paper, we intend to show that essentially all p-algebras split by E are obtained by a natural composition of two constituents: a certain purely inseparable field t containing E and any abelian normal extension N of K whose Galois group is related, in a manner to be described, to the structure of E. We must dwell a little on the nature of these ingredients. Consider a subgroup X of the multiplicative group E* of E such that X contains the multiplicative group K* of K. Such a group will be called regular, if any system of representatives of X modulo K* is linearly independent over K. E itself is called regular if it is additively generated by some regular subgroup of E*, which in this case will be called a maximal regular subgroup. Just below the corollary for Theorem 2 in [2], it was shown that every finite purely inseparable extension E can be further extended to a regular one A with the same exponent over K and also finite. In what follows, we require E to be regular. The field originally given may have to be enlarged to fulfill this condition, just as a separable field is extended to a normal one in the theory of crossed products. We assume, therefore that P=E. It follows at once from Theorem 1 of [2] that the group G(X) =X/K* associated with a maximal regular subgroup X of E* is independent of X. There is thus a unique p-group G attached to E. The group X is an extension of K* by G. Hence with the selection of a maximal regular group X we obtain a cohomology class ;gEH2(G, K*). For the sequel let X be fixed. As for N, it will be a normal extension of F with Galois group rPG. However, N need not be a field; in general it may be a direct sum of fields

Constructible units in abelian p-group rings

Let A be a finite abelian group of exponent pm>1, an odd prime power, and consider the group Ω(A) of units in the group ring ZA which appear circular in each cyclotomic Wedderburn component. We explicitly construct generators for a subgroup Λ(A)⊆Ω(A) of p-power index, which equals Ω(A) if and only if p is regular. Moreover, we determine the structure of Λ(A) as a G-module, where G is the unit of Z mod pmZ.

Units in regular abelian p-group rings

An element of the group ring RCA], of some finite abelian group A of odd order over a commutative ring R, will be called symmetric, if it is left fixed by the involution on RCA] induced by u + u-l (SEA). One of the basic facts about units in Z[A] is that U,Z[A] =AU,Z[A] and U,Z[A] is torsion-free (cf. [2, Lemma 2.6]), where generally U, denotes the units of augmentation one, and 17, the subgroup of symmetric elements therein. In this note, we investigate the images of three natural maps for abelian p-groups, A (p odd):

Units and class-groups in integral elementary abelian group rings

Let A be an elementary abelian p-group, p odd and semi-regular. In the group-ring ZA consider units which lie in ZC for some cyclic C ⊂ A, and let m(A) be the index of the subgroup generated by all of these in the full unit-group. Main result: m(A) = 1 iff p is regular. Moreover m(A) is related to a certain group of ideal classes in ZA.

Local units and circular index in abelian p-group rings

Abstract Let A be a finite abelian group of exponent p m >1 , an odd prime power, and consider the Z > p -module Δ + p ( A ) generated by the set {( z −1)+( z -1 −1)∣ z ∈ A } in the group ring Z p A . We show that the multiplicative group S p ( A )=1+ Δ + p ( A ) has a Z p -basis of the form { w p ( z )∣1≠ z ∈ A }, where w p ( x ) is a fixed Z p -polynomial. The integral unit group S(A)=( Z A) x ∩S p (A ) is known to have a canonical subgroup Λ( A ) of a similar structure; it lies in the group Ω( A ) of all units appearing circular under every character of A . Comparing the two situations, we conclude that the index [Ω( A ) : Λ( A )] equals the order of a certain group of ideal classes of Z A .

Kummer's lemma for cyclic 2-groups

AbstractFor cyclic 2-groupsC, we characterize the kernel of the map

Cyclotomic units over finite fields

U\mathbb{Z}C \to U\mathbb{F}_2 F

A cohomological characterization of finite Nilpotent groups

induced on the units of the integral group ring by the coefficient reduction