Unramified Quaternion Extensions of Quadratic Number Fields

book describes a constructive approach to the inverse Galois problem: Given a finite group G and a field K, determine whether there exists a Galois extension of K whose Galois group is isomorphic to G. Further, if there is such a Galois extension, find an explicit polynomial over K whose Galois group is the prescribed group G. The main theme of the book is an exposition of a family of " generic " poly-nomials for certain finite groups, which give all Galois extensions having the required group as their Galois group. The existence of such generic polyno-mials is discussed, and where they do exist, a detailed treatment of their construction is given. The book also introduces the notion of " generic dimension " to address the problem of the smallest number of parameters required by a generic polynomial. published by the press syndicate of the university of cambridge A catalogue record for this book is available from the British Library.

Generic Polynomials Constructive Aspects Of The Inverse Galois Problem

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.

/pdf/automatic-realizability-of-galois-groups-of-order-16-4og868g55d.pdf

Automatic realizability of Galois groups of order 16

The monodromy method—featuring braid group action—first appeared as a moduli space approach for finding solutions of arithmetic problems that produce reducible variables separated curves. Examples in this paper illustrate its most interesting aspect: investigating the moduli space of exceptions to a specific diophantine outcome. Explicit versions of Hilbert’s irreducibility theorem and Davenport’s problem fostered this technique and motivated the genus 0 problem started by J. Thompson and taken up by many group theorists. We review progress on the genus 0 problem in 0 characteristic, and its quite different contributions in positive characteristic. Example: Let f and h be polynomials with coefficients in a number field K. The classification of finite simple groups shows there is a bound on exceptional degrees for f to the following result. If f is indecomposable and h is not a composition with f , then f(x) − h(y) is irreducible. This answered challenge problems on factorization of variables separated polynomials posed by A. Schinzel in the early 60’s. This limitation result holds, however, only in characteristic 0, one difference between the Genus 0 Problem here and in positive characteristic. Finite field example—Davenport’s Problem: For each finite field Fq there are infinitely many surprising polynomial pairs (f, h) (of degree prime to the characteristic) whose value sets are equal over Fqt , t = 1, 2, . . . . Though we include unpublished results from 30 years ago, a new set of problems stretch the methods. Example of a general theme: Let Md be the elements of Q of degree no more than d over Q. The degree d reducibility set of f , Rf (d) is Rf (d) = {z0 ∈ Md | f(y) − z0 is reducible over Q(z0)}. Similarly, there is a value set Vf (d) (degree 1 fiber). Following a reduction due to G. Frey, for any integer d, there are polynomials f of unbounded degree satisfying Rf (d) \ Vf (d) is finite. The full monodromy method finds precise arithmetic information by extending observations from modular curves. A semi-classical observation: Divisors with support in cusp points on modular curves generate a torsion group on the Jacobian of the curve. Illustrations here (from alternating group covers) show generalizations of this issue are ubiquitous with the monodromy method. Date: October 25, 2001. 1991 Mathematics Subject Classification. Primary 12F05; Secondary 14H40, 14D20, 14E20.

Variables separated polynomials, the genus 0 problem and moduli spaces

Parameterizing solutions to any Galois embedding problem over Z/pnZ with elementary p-abelian kernel

On $2$-groups as Galois groups

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On $2$-groups as Galois groups