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2006 | Buch

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Field Theory

verfasst von: Steven Roman

Verlag: Springer New York

Buchreihe : Graduate Texts in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Intended for graduate courses or for independent study, this book presents the basic theory of fields. The first part begins with a discussion of polynomials over a ring, the division algorithm, irreducibility, field extensions, and embeddings. The second part is devoted to Galois theory. The third part of the book treats the theory of binomials. The book concludes with a chapter on families of binomials – the Kummer theory.

This new edition has been completely rewritten in order to improve the pedagogy and to make the text more accessible to graduate students. The exercises have also been improved and a new chapter on ordered fields has been included.

About the first edition:

" ...the author has gotten across many important ideas and results. This book should not only work well as a textbook for a beginning graduate course in field theory, but also for a student who wishes to take a field theory course as independent study."

-J.N. Mordeson, Zentralblatt

"The book is written in a clear and explanatory style. It contains over 235 exercises which provide a challenge to the reader. The book is recommended for a graduate course in field theory as well as for independent study."

- T. Albu, MathSciNet

Inhaltsverzeichnis

Frontmatter

Preliminaries

Chapter 0. Preliminaries

Abstract

The purpose of this chapter is to review some basic facts that will be needed in the book. The discussion is not intended to be complete, nor are all proofs supplied. We suggest that the reader quickly skim this chapter (or skip it altogether) and use it as a reference if needed.

Field Extensions

Frontmatter

Chapter 1. Polynomials

Abstract

In this chapter, we discuss properties of polynomials that will be needed in the sequel. Since we assume that the reader is familiar with the basic properties of polynomials, some of the present material may constitute a review.

Chapter 2. Field Extensions

Abstract

In this chapter, we will describe several types of field extensions and study their basic properties.

Chapter 3. Embeddings and Separability

Abstract

Let us recall a few facts about separable polynomials from Chapter 1.

Chapter 4. Algebraic Independence

Abstract

In this chapter, we discuss the structure of an arbitrary field extension F < E. We will see that for any extension F < E. there exists an intermediate field F < F(S) < E whose upper step F(S) < E is algebraic and whose lower step F < F(S) is purely transcendental, that is, there is no nontrivial polynomial dependency (over F) among the elements of S, and so these elements act as “independent variables” over F. Thus, F(S) is the field of all rational functions in these variables.

Galois Theory

Frontmatter

Chapter 5. Galois Theory I: An Historical Perspective

Abstract

Galois theory sits atop a structure of work began about 4000 years ago on the question of how to solve polynomial equations algebraically by radicals, that is, how to solve equations of the form

$$ a_n x^n + a_{n - 1} x^{n - 1} + ... + a_0 = 0 $$

by applying the four basic arithmetical operations (addition, subtraction, multiplication and division), and the taking of roots, to the coefficients of the equation and to other “known” quantities (such as elements of the base field).

Chapter 6. Galois Theory II: The Theory

Abstract

The traditional Galois correspondence between intermediate fields of an extension and subgroups of the Galois group is one of the main themes of this book. We choose to approach this theme through a more general concept, however.

Chapter 7. Galois Theory III: The Galois Group of a Polynomial

Abstract

In this chapter, we pass from the highly theoretical material of the previous chapter to the somewhat more concrete, where we apply the results of the previous chapter to some special Galois correspondences.

Chapter 8. A Field Extension as a Vector Space

Abstract

In this chapter, we take a closer look at a finite extension F < E from the point of view that E is a vector space over F. It is clear, for instance, that any σ ∈ G_F(E) is a linear operator on E over F. However, there are many linear operators that are not field automorphisms. One of the most important is multiplication by a fixed element of E, which we study next.

Chapter 9. Finite Fields I: Basic Properties

Abstract

In this chapter and the next, we study finite fields, which play an important role in the applications of field theory, especially to coding theory, cryptology and combinatorics. For a thorough treatment of finite fields, the reader should consult the book Introduction to Finite Fields and Their Applications, by Lidl and Niederreiter, Cambridge University Press, 1986.

Chapter 10. Finite Fields II: Additional Properties

Abstract

There are various ways in which to represent the elements of a finite field. Since every finite field F is simple, it has the form F = GF(p)(α for some α ∈ F and so the elements of F are polynomials in α of degree less than deg (α). Another way to represent the elements of a finite field is to use the fact that GF(q)* is cyclic, and so its elements are all powers of a group primitive element.

Chapter 11. The Roots of Unity

Abstract

Polynomials of the form xⁿ − u, where 0 ≠ u ∈ F, are known as binomials. Even though binomials have a simple form, their study is quite involved, as is evidenced by the fact that the Galois group of a binomial is often nonabelian. As we will see, an understanding of the binomial xⁿ − 1 is key to an understanding of all binomials.

Chapter 12. Cyclic Extensions

Abstract

Continuing our discussion of binomials begun in the previous chapter, we will show that if S is a splitting field for the binomial xⁿ − u, then S = F(ω, α) where ω is a primitive nth root of unity. In the tower

$$ F < F\left( \omega \right) < F\left( {\omega ,\alpha } \right) $$

the first step is a cyclotomic extension, which, as we have seen, is abelian and may be cyclic. In this chapter, we will see that the second step is cyclic of degree. d | n and α can be chosen so that min(α, F(ω)) = x^d − α^d. Nevertheless, as we will see in the next chapter, the Galois group G_F(S need not even be abelian.

Chapter 13. Solvable Extensions

Abstract

We now turn to the question of when an arbitrary polynomial equation p(x) = 0 is solvable by radicals. Loosely speaking, this means (for char(F) = 0) that we can reach the roots of p(x) by a finite process of adjoining nth roots of existing elements, that is, by a finite process of passing from a field K to a field K(α), where α is a root of a binomial xⁿ − u, with u ∈ K. We begin with some basic facts about solvable groups.

The Theory of Binomials

Frontmatter

Chapter 14. Binomials

Abstact

We continue our study of binomials by determining conditions that characterize irreducibility and describing the Galois group of a xⁿ − u binomial in terms of 2 × 2 matrices over ℤ_n. We then consider an application of binomials to determining the irrationality of linear combinations of radicals. Specifically, we prove that if p₁,....,p_m are distinct prime numbers, then the degree of

$$ \mathbb{Q}\left( {\sqrt[n]{{p_1 }},...,\sqrt[n]{{p_m }}} \right) $$

over ℚ is as large as possible, namely, n^m. This implies that the set of all products of the form

$$ \sqrt[n]{{p_1^{e\left( 1 \right)} }}...\sqrt[n]{{p_m^{e\left( m \right)} }} $$

where 0 ≤ e(i) ≤ n − 1, is linearly independent over ℚ For instance, the numbers

$$ 1,\sqrt[4]{3} = \sqrt[{60}]{{3^{15} }},\sqrt[5]{4} = \sqrt[{60}]{{2^{24} }}{\mathbf{ }}{\text{and}}{\mathbf{ }}\sqrt[6]{{72}} = \sqrt[{60}]{{2^{30} 3^{20} }} $$

are of this form, where p₁ = 2, p₂ = 3. Hence, any expression of the form

$$ a_1 \sqrt[4]{3} + a_2 \sqrt[5]{4} + a_3 \sqrt[6]{{72}} $$

where a_i ∈ ℚ, must be irrational, unless a_i = 0 for all i.

Chapter 15. Families of Binomials

Abstract

In this chapter, we look briefly at families of binomials and their splitting fields and Galois groups. We have seen that when the base field F contains a primitive nth root of unity, cyclic extensions of degree d | n correspond to splitting fields of a single binomial xⁿ − u. More generally, we will see that abelian extensions correspond to splitting fields of families of binomials. We will also address the issue of when two families of binomials have the same splitting field.

Backmatter

Titel: Field Theory
verfasst von: Steven Roman
Verlag: Springer New York
Electronic ISBN: 978-0-387-27678-6
Print ISBN: 978-0-387-27677-9
DOI: https://doi.org/10.1007/0-387-27678-5