Tara L. Smith

What It Is They Do: Differentiating Knowledge and Literacy Practices Across Content Disciplines

This study addresses the gap that exists between literacy educators’ knowledge of content disciplines and the literacy strategies often suggested for use in content classrooms. The authors worked with disciplinary experts in mathematics and geography to understand the differences that exist in their conceptions of the disciplines and what it means to be literate in them. Study results were categorized into themes, which include “major understandings of the field,” “literacy in the discipline,” and “practice of the discipline.” Further subcategories include texts used in the discipline; the types of strategies used to become literate in the discipline, and the types of questions addressed within the discipline. Implications for teachers and teacher candidates are also addressed.

Automatic realizability of Galois groups of order 16

In this article we are concerned with automatic realizability results for small Galois 2-groups, particularly those of order 16. We ask when it is true that the realizability of a particular group G as a Galois group over a field K implies the realizability of another group H as a Galois group over K. In some cases, these results hold over all fields. In others, the automatic realizability depends on specific properties of K. We are also interested in field-theoretic criteria which guarantee the realizability of certain groups. By a famous result of Witt [Wi:1936], Satz, p.237, the realizability of a 2-group G over a field of characteristic 2 depends only on the minimal number of generators of G. For this reason we will always assume our fields to be of characteristic not 2. We obtain our results primarily by considering the obstructions to the realizability of the groups, expressed in terms of products of quaternion algebras in the Brauer group Br(K) of the base field K. By manipulating these expressions, one can often show that triviality of the obstruction for one group implies triviality of the obstruction for another. This is particularly true if additional information on the field, such as its level, is taken into consideration. The obstructions for these groups have appeared in various references. A complete survey of known results is provided in [GSS:1995]. We quote these results below and use them heavily in this article. Realizability questions have been considered previously in the two articles by C. U. Jensen, [Je:1989] and [Je:1992]. Jensen considers the structure of the groups explicitly in most instances, and often describes precisely how to construct fields realizing the groups, whereas our approach is less constructive. He also works with finite 2-groups in general, whereas we work just with groups of order at most 16. We obtain a number of new realizability results by our methods. Our goal is to systematically examine the groups of order 16, and what can be said about the realizability of each group as a Galois group over fields of characteristic not 2.

Extra-special groups of order

In this article we examine conditions for the appearance or nonappearance of the two extra-special 2-groups of order 32 as Galois groups over a field F of characteristic not 2. The groups in question are the central products DD of two dihedral groups of order 8, and DQ of a dihedral group with the quaternion group, obtained by identifying the central elements of order 2 in each factor group. It is shown that the realizability of each of these groups as Galois groups over F implies the realizability of other 2-groups (which are not their quotient groups), and in turn that realizability of certain other 2-groups implies the realizability of DD and DQ. We conclude by providing an explicit construction of field extensions with Galois group DD.

32

The intriguing relation between the theory of quadratic forms and Galois theory has been of interest for a long time. (See for example [Wi:1936], [Wr:1979], [Wr:1983], [Wr:1985], [JWr:1989], [AEJ:1984], among others.) However, recently the connection between the Witt ring structure for a field F (of characteristic not 2) and its Galois groups was made quite precise. If L is a Galois field extension of F , with Gal(L/F ) ∼= G, we call L a G-extension of F . Given a field F , with charF 6= 2, one can consider the field extension F /F which is the compositum over F of all Z/2Z, Z/4Z-, and D4-extensions of F . (Here D4 denotes the dihedral group of order 8.) One can then show that the Galois group GF of F (3) over F , hereafter referred to as the W-group of F , is determined by W (F ), and that GF determines W (F ) except in the case when the level s(F ) of F is ≤ 2 and the form 〈1, 1〉 is universal. (In this paper basic knowledge of quadratic form theory and profinite groups will be assumed. See [La:1973] or [Sc:1985] for the former and [N:1971] and [Se:1965] for the latter. Throughout we will assume all fields to be of characteristic not 2.) In other words, knowledge of GF is essentially equivalent to knowledge of W (F ). (See [MiSm:1993], [:1990], [MiSp:1995], [Sm:1988], [Sp:1987].) This relationship between a specific Galois group of F and W (F ) opened a new way of attacking some questions in quadratic from theory and posed other new questions. In particular, it allows one to use the techniques of inverse Galois theory to study some classical problems in the theory of Witt rings. One of the most outstanding problems is the characterization of Witt rings in the category of all rings. In spite of many efforts, very little is known. Indeed, we know only

as Galois groups

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.

Decomposition of Witt rings and Galois groups

To each eld F of characteristic not 2, one can associate a certain Galois group GF, the socalled W-group of F, which carries essentially the same information as the Witt ring W(F )o f equivalence classes of anisotropic quadratic forms over F. There is a close connection between (nontrivial) involutions in GF 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. c 2000 Elsevier Science B.V. All rights reserved. MSC: 11E81; 12D15

Realizability and automatic realizability of Galois groups of order 32

This article examines the realizability of groups of order 64 as Galois groups over arbitrary fields. Specifically, we provide necessary and sufficient conditions for the realizability of 134 of the 200 noncyclic groups of order 64 that are not direct products of smaller groups.

Formally real fields from a Galois theoretic perspective

A b st r ac t . To each field F of characteristic not 2, one can associate a certain Galois group GF , the so-called W-group of F , which carries essentially the same information as the Witt ring W (F ) of F . In this paper we investigate the connection between GF and GF (√a), where F ( √ a) is a proper quadratic extension of F . We obtain a precise description in the case when F is a pythagorean formally real field and a = −1, and show that the W-group of a proper field extension K/F is a subgroup of the W-group of F if and only if F is a formally real pythagorean field and K = F ( √ −1). This theorem can be viewed as an analogue of the classical Artin-Schreier’s theorem describing fields fixed by finite subgroups of absolute Galois groups. We also obtain precise results in the case when a is a double-rigid element in F . Some of these results carry over to the general setting.

Galois realizability of groups of order 64

Let F be a field of characteristic other than 2. Let F (2) denote the compositum over F of all quadratic extensions of F , let F (3) denote the compositum over F (2) of all quadratic extensions of F (2) that are Galois over F , and let F{3} denote the compositum over F (2) of all quadratic extensions of F (2). This paper shows that F (3) = F{3} if and only if F is a rigid field, and that F (3) = K(3) for some extension K of F if and only if F is Pythagorean and K = F( √−1). The proofs depend mainly on the behavior of quadratic forms over quadratic extensions, and the corresponding norm maps.

W-Groups under Quadratic Extensions of Fields

Abstract We investigate the connections between the values assumed by binary quadratic forms over a field F (of characteristic not 2) and certain 2-groups arising as Galois groups over F. The groups in question will always be quotients of the so-called W-group of F. This group is the Galois group of the compositum over F of all quadratic extensions, cyclic extensions of order 4, and dihedral extensions of order 8. We show how the W-group and its quotients determine the values assumed by any binary form. The main result is to apply these ideas to give a simple characterization via Galois groups of fields with level ≤4.