Book contents
- Frontmatter
- Contents
- Editor's Statement
- Section Editor's Foreword
- Preface
- Historical Introduction
- Prerequisites
- Notation
- Field Extensions and Galois Theory
- Chapter 1 Preliminaries on Fields and Polynomials
- Chapter 2 Algebraic Extensions
- Chapter 3 Galois Theory
- Chapter 4 Transcendental Extensions
- References and Selected Bibliography
- Index
Chapter 1 - Preliminaries on Fields and Polynomials
Published online by Cambridge University Press: 05 June 2013
- Frontmatter
- Contents
- Editor's Statement
- Section Editor's Foreword
- Preface
- Historical Introduction
- Prerequisites
- Notation
- Field Extensions and Galois Theory
- Chapter 1 Preliminaries on Fields and Polynomials
- Chapter 2 Algebraic Extensions
- Chapter 3 Galois Theory
- Chapter 4 Transcendental Extensions
- References and Selected Bibliography
- Index
Summary
FIELDS OF FRACTIONS
A basic relationship between the field Q and its subdomain Z with which the reader is already familiar is that every element of Q can be expressed as a fraction with numerator and denominator in Z. It is clear that such a connection can be meaningfully formulated in the more general context of an arbitrary field and a subdomain. This leads to the general concept of a field of fractions, which we shall discuss in this section.
Let A be a domain. By a field of fractions ofA we understand a field K having A as a subdomain and such that every element of K is expressible in the form α/β with α, β ∈ A and β ≠ 0.
It follows that if A is a domain, and if K is a field of fractions of A, then no proper subfield of K contains A, and K is a field of fraction of every intermediate domain between A and K.
In particular, a field is its only field of fractions.
The preceding definition immediately suggests the questions of existence and essential uniqueness of fields of fractions of domains. In the discussion that follows we shall see that these can be settled completely.
If a domain is given as a subdomain of a field, there is no difficulty in showing that it admits a field of fractions.
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- Field Extensions and Galois Theory , pp. 1 - 40Publisher: Cambridge University PressPrint publication year: 1984