Helen G. Grundman

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.

Realizability and automatic realizability of Galois groups of order 32

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.

Galois realizability of groups of order 64

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.

EXPLICIT RESOLUTIONS OF CUBIC CUSP SINGULARITIES

Resolutions of cusp singularities are crucial to many techniques in computational number theory, and therefore finding explicit resolutions of these singularities has been the focus of a great deal of research. This paper presents an implementation of a sequence of algorithms leading to explicit resolutions of cusp singularities arising from totally real cubic number fields. As an example, the implementation is used to compute values of partial zeta functions associated to these cusps.

ℚ of galois 2-extensions

The ℚ of any finite Galois 2-extension of ℚ is shown to depend only on the ℚ of its maximal elementary abelian intermediate field, which must be either quadratic (and hence always ℚ-adequate) or biquadratic over ℚ. A precise description of those biquadratic extensions of ℚ which are ℚ-adequate is given. This then gives a method for explicitly determining whether any given finite Galois 2-extension of ℚ can arise as a subfield of a ℚ-central division algebra.

Fixed points of augmented generalized happy functions

An augmented generalized happy function

Augmented generalized happy functions

S_{[c,b]}

On the Diophantine equation X^{2N}+2^{2\alpha }5^{2\beta }{p}^{2\gamma } = Z^5

maps a positive integer to the sum of the squares of its base

On the Diophantine equation NX2+2L3M=YN

b

On the Diophantine equation

digits plus