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In this presentation of the Galois correspondence, modern theories of groups and fields are used to study problems, some of which date back to the ancient Greeks. The techniques used to solve these problems, rather than the solutions themselves, are of primary importance. The ancient Greeks were concerned with constructibility problems. For example, they tried to determine if it was possible, using straightedge and compass alone, to perform any of the following tasks? (1) Double an arbitrary cube; in particular, construct a cube with volume twice that of the unit cube. (2) Trisect an arbitrary angle. (3) Square an arbitrary circle; in particular, construct a square with area 1r. (4) Construct a regular polygon with n sides for n > 2. If we define a real number c to be constructible if, and only if, the point (c, 0) can be constructed starting with the points (0,0) and (1,0), then we may show that the set of constructible numbers is a subfield of the field R of real numbers containing the field Q of rational numbers. Such a subfield is called an intermediate field of Rover Q. We may thus gain insight into the constructibility problems by studying intermediate fields of Rover Q. In chapter 4 we will show that (1) through (3) are not possible and we will determine necessary and sufficient conditions that the integer n must satisfy in order that a regular polygon with n sides be constructible.

Inhaltsverzeichnis

Frontmatter

Chapter I. Preliminaries — Groups and Rings

Abstract
In this chapter we present the background required in the study of the Galois correspondence. We give the basic definitions and theorems of the elementary theory of groups and rings, concentrating on examples that will be used in later chapters. Although some of the more straightforward proofs are left as exercises, the majority of the proofs in the first two sections are presented fully as we guide the student through the process of studying groups via their normal subgroups and quotient groups.
Maureen H. Fenrick

Chapter II. Field Extensions

Abstract
The field Q of rationals is a subfield of the field R of reals, which is, in turn, a subfield of the field C of complex numbers. We then write Q≺R≺C and say that R is an intermediate field of the extension C over Q.
Maureen H. Fenrick

Chapter III. The Galois Correspondence

Abstract
In this chapter, we will study the group G of K-automorphisms of an extension field F of K. In particular, we will show that, if F is a finite, normal extension of K and K has characteristic 0, then o (G) = [F : K] and there is a one-to-one, order reversing correspondence between the set of intermediate fields of the extension K≺F and the set of subgroups of G. We will then show that this correspondence also preserves normality.
Maureen H. Fenrick

Chapter IV. Applications

Abstract
In this chapter we present some of the diverse applications of the Galois correspondence.
Maureen H. Fenrick

Backmatter

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