Ž. M. Marshall 2000, Comm. Algebra 28, 1157᎐2273 has generalized the notion of )-ordering to the setting of a ring with involution. In this paper we analyze the Ž . ways in which a given )-ordering on the set of symmetric elements can be extended to a multiplicatively closed ordering on a larger set of elements. A complete answer is given for Ore domains. ᮊ

To each ÿeld F of characteristic not 2, one can associate a certain Galois group G F , the socalled W-group of F, which carries essentially the same information as the Witt ring W (F) of equivalence classes of anisotropic quadratic forms over F. There is a close connection between (nontrivial) involutions in G F and orderings on F. The purpose of this paper is to investigate how the lattice of orderings and preorderings on F is determined by GF , and to provide a Galoistheoretic version of reduced Witt rings.

This article provides necessary and sufficient conditions for each group of order 32 to be realizable as a Galois group over an arbitrary field. These conditions, given in terms of the number of square classes of the field and the triviality of specific elements in related Brauer groups, are used to derive a variety of automatic realizability results.

Abstract. This article examines the realizability of small groups of order 2 k , k ≤ 4, as Galois groups over arbitrary fields of characteristic not 2. In particular we consider automatic realizability of certain groups given the realizability of others.

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