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2018 | Buch

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Galois Theory Through Exercises

verfasst von: Prof. Juliusz Brzeziński

Verlag: Springer International Publishing

Buchreihe : Springer Undergraduate Mathematics Series

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

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Über dieses Buch

This textbook offers a unique introduction to classical Galois theory through many concrete examples and exercises of varying difficulty (including computer-assisted exercises).

In addition to covering standard material, the book explores topics related to classical problems such as Galois’ theorem on solvable groups of polynomial equations of prime degrees, Nagell's proof of non-solvability by radicals of quintic equations, Tschirnhausen's transformations, lunes of Hippocrates, and Galois' resolvents. Topics related to open conjectures are also discussed, including exercises related to the inverse Galois problem and cyclotomic fields. The author presents proofs of theorems, historical comments and useful references alongside the exercises, providing readers with a well-rounded introduction to the subject and a gateway to further reading.

A valuable reference and a rich source of exercises with sample solutions, this book will be useful to both students and lecturers. Its original concept makes it particularly suitable for self-study.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Solving Algebraic Equations

Abstract

The aim of this chapter is to show how equations of degrees less than 5 can be solved. We highlight well-known formulae for the quadratic equation and show how to find similar formulae for cubic and quartic equations. We also explain why as early as the eighteenth century mathematicians started to doubt the possibility to find solutions for general quintic equations (or equations of higher degrees) using the four arithmetic operations and extracting roots applied to coefficients. We give examples of quantic equations for which such formulae exist (e.g. de Moivre’s quintics) and show that the ideas which work for equations of degrees up to 4 have no evident generalizations. We also briefly discuss “casus irreducibilis” related to cubic equations.

Juliusz Brzeziński

Chapter 2. Field Extensions

Abstract

In this chapter, we recall some general notations and properties of fields (incl. the notion of characteristic) which are important to Galois theory but are not always sufficiently stressed in introductory courses on basic algebraic structures. Importantly, there are different notations for fields, which we fix in this chapter for the remainder of the book.

Juliusz Brzeziński

Chapter 3. Polynomials and Irreducibility

Abstract

In this chapter, we present facts on zeros of polynomials and discuss some basic methods to decide whether a polynomial is irreducible or reducible, including Gauss’ lemma, the reduction of polynomials modulo prime numbers ((irreducibility over finite fields), and Eisenstein’s criterion.

Juliusz Brzeziński

Chapter 4. Algebraic Extensions

Abstract

This chapter describes a kind of field extension most common in Galois theory: algebraic extensions, i.e. extensions K ⊆ L in which each element α ∈ L is a zero of a nontrivial polynomial with coefficients in K. We relate the elements of algebraic extensions to the corresponding polynomials and look at the structures of the simplest extensions K(α) of K. We also introduce the notion of the degree of a field extension and prove some of its properties such as the Tower Law.

Juliusz Brzeziński

Chapter 5. Splitting Fields

Abstract

We consider field extensions which are generated by all zeros of a given polynomial over a field containing its coefficients. Such a field is called a splitting field. Splitting fields of polynomials play a central role in Galois theory. It is intuitively clear that a splitting field of a polynomial is a very natural object carrying a lot of information about it. We will find a confirmation of this in the following chapters, in particular, when we come to the main theorems of Galois theory (Chap. 9) and its applications in subsequent chapters (e.g. in Chap. 13, where we study the solvability of equations by radicals). As an important example, we study finite fields in this chapter. These can be easily described as splitting fields of very simple polynomials over finite prime fields. We further consider the notion of an algebraic closure of a field K, which is a minimal field extension of K, containing a splitting field of every polynomial with coefficients in K.

Juliusz Brzeziński

Chapter 6. Automorphism Groups of Fields

Abstract

In this chapter, we study automorphism groups of fields and introduce Galois groups of finite field extensions. The term “Galois group” is often reserved for automorphism groups of Galois field extensions, which we define and study in Chap. 9. The terminology used in this book is very common and has several advantages in textbooks (i.e. it is easier to formulate exercises). A central result of this chapter is Artin’s lemma, which is a key result in the modern presentation of Galois theory. In the exercises, we find Galois groups of many field extensions and we use also use this theorem for various problems on field extensions and their automorphism groups.

Juliusz Brzeziński

Chapter 7. Normal Extensions

Abstract

In this chapter, we look at the splitting fields of polynomials and emphasize one important property of such fields. This property is contained in the definition of a normal extension: An extension of a field is normal if every irreducible polynomial over this field with one zero in the extension has already all its zeros in it. This means that the extension contains a splitting field of any irreducible polynomial which has at least one zero in it. This definition works equally well for any field extension (also infinite), but we focus here on finite extensions and find that normal extensions and splitting fields of polynomials form exactly the same class. We further discuss a normal closure of a finite field extension. Galois extensions are those which are normal and separable. The separable extensions are discussed in the next chapter.

Juliusz Brzeziński

Chapter 8. Separable Extensions

Abstract

In this chapter, we discuss the last property of field extensions which is needed to define Galois extensions: separability (which is usually satisfied in most common situations). An extension of a field is separable if any irreducible polynomial with coefficients in this field does not have multiple zeros. All extensions of fields of characteristic zero and all finite extensions of finite fields have this property. For this reason, there are sometimes “simplified presentations” of Galois theory in which one studies only fields of characteristic zero and finite fields. It is then not necessary to mention separability and the theoretical background necessary for the main theorems of Galois theory is more modest. We choose to discuss separability here as nonseparable field extensions are important in many branches of mathematics. In this chapter, we characterize separable extensions and prove the theorem on primitive element which says that a finite separable extension can be generated over its ground field by only one element. This theorem is usually part of any standard course on the subject.

Juliusz Brzeziński

Chapter 9. Galois Extensions

Abstract

In this chapter, we arrive at the main theorems of Galois theory. Following the material covered in previous chapters, we are now equipped to define Galois extensions, i.e. the extensions which are both normal and separable. One of the most central results is Galois’ correspondence between the subgroups of the Galois group of such an extension and the intermediate subfields of it. In the exercises, we find many examples and interesting properties of Galois extensions. Several exercises are related to the inverse problem in Galois theory: to construct a Galois extension of the rational numbers with given group as its Galois group. In a series of exercises, this problem is solved for different groups of small orders and for all cyclic groups using a special case of Dirichlet’s theorem on primes in arithmetic progression, which is proved in the next chapter.

Juliusz Brzeziński

Chapter 10. Cyclotomic Extensions

Abstract

This chapter illustrates the general theory of Galois extensions in a special case. We study field extensions, mostly of the rational numbers, generated by the roots of 1. Even if such fields are simple to describe in purely algebraic terms, they are rich as mathematical objects. We explore some of their properties, which find different applications in number theory and algebra. Among many applications, there is a proof of a special case of Dirichlet’s theorem on primes in arithmetic progression using the cyclotomic polynomials. This chapter also includes an exercise on a solution of the inverse Galois problem for all abelian groups.

Juliusz Brzeziński

Chapter 11. Galois Modules

Abstract

This chapter also illustrates the main theorems of Galois theory, though with more of a theoretical focus. We discuss some general notions and prove three important theorems that have several applications in algebra and number theory. These three theorems are related in one way or another to Galois modules, i.e. groups on which Galois groups of field extensions act as transformation groups. The first theorem is on the existence of so-called normal bases and concerns especially nice bases of finite Galois extensions. It is related to the Galois module structure on the additive group of the field. The second theorem, Hilbert’s theorem 90, is related to both additive and multiplicative structures of Galois extensions. This theorem has numerous applications which are discussed in the exercises. The third theorem is related to Hilbert’s theorem 90. Here we discuss so-called Kummer extensions. A particular case of such extensions are cyclic extensions. Hilbert’s theorem 90 may be considered in the context of so-called cohomology groups (defined for G-modules), which we mention briefly. We further classify all cubic and quartic Galois extensions of rational numbers in the context of cohomology groups.

Juliusz Brzeziński

Chapter 12. Solvable Groups

Abstract

In this chapter, we define solvable groups, discuss several examples of such groups and prove some of their properties which are needed later on. The background on solvable groups presented in this chapter will become relevant in subsequent chapters when we discuss the solvability of equations by radicals—one of the central applications of Galois theory presented in this book.

Juliusz Brzeziński

Chapter 13. Solvability of Equations

Abstract

In this chapter, we show that equations solvable by radicals are characterized by the solvability of their Galois groups. This immediately implies that general equations of degree 5 and above are not solvable by radicals. If one has a more modest goal to prove that the fifth degree general equation over a number field is not solvable by radicals, then there exists a simple argument by Nagell which only requires limited knowledge of field extensions and no knowledge of Galois theory. We consider Nagell’s proof in the exercises. This chapter further outlines Weber’s theorem on irreducible equations of prime degree (at least 5) with only two nonreal zeros, which are examples of non-solvable equations. We further discuss Galois’ classical theorem, which gives a characterization of irreducible solvable polynomials of prime degree. Both Galois’ and Weber’s results give examples of concrete unsolvable polynomials over the rational numbers. The solvability by real radicals in connection with “casus irreducibilis” is also discussed.

Juliusz Brzeziński

Chapter 14. Geometric Constructions

Abstract

In this chapter, we discuss some straightedge-and-compass constructions. We are particularly interested in the following classical problems: the impossibility of squaring the circle, doubling the cube and angle trisection. The problems of impossibility (or possibility) of some geometric constructions by using different means (such as straightedge-and-compass, only compass, or other means) are usually discussed in courses on Galois theory. Meanwhile, the classical impossibility problems mentioned above do not require extensive knowledge on Galois groups (if any at all); only some basic knowledge of finite field extensions is required. Other problems, such as Gauss’ theorem on straightedge-and-compass constructible regular polygons require further knowledge related to Galois groups of field extensions. This chapter contains several exercises concerned with geometric straightedge-and-compass constructions. We prove two theorems: the first tends to be used in proofs of impossibility of some straightedge-and-compass constructions; the second tends to be used in proofs of possibility.

Juliusz Brzeziński

Chapter 15. Computing Galois Groups

Abstract

In this chapter, we discuss computations of Galois groups. In general, computing the Galois group of a given polynomial is numerically complicated when the degree of the polynomial is modestly high. The numerical methods depend on the knowledge of transitive subgroups of the symmetric groups. Here, we discuss some theoretical background on numerical methods (which are implemented in some computer packages) and apply it in a few cases. In the exercises, we illustrate how to compute and classify Galois groups for low degree polynomials by specifying some numerical invariants, which provides information on the isomorphism type of the Galois group depending on their values. We do this for polynomials of degrees 3 and 4. We further discuss the Galois resolvents and use them to proof a general theorem by Richard Dedekind, which relates the Galois group of an integer irreducible polynomial to Galois groups of its reductions modulo prime numbers. Several exercises are concerned with Dedekind’s theorem, allowing for the construction of polynomials with given Galois groups and the solution of the inverse problem for the symmetric group S_n.

Juliusz Brzeziński

Chapter 16. Supplementary Problems

Abstract

In this chapter, we present 100 problems without solutions (but occasionally with some hints, comments or references). These are original, interesting or challenging problems covering various aspects of Galois theory. Several of these problems can be used as a starting point for student projects, such as problems related to the normal core of groups, Galois index, the notion of elements of field extensions essentially defined over a subfield. Some of the problems are suitably structured in order to introduce some interesting topics that are typically not covered in standard texts on the subject, incl. Dedekind’s duality, Tschirnhausen’s transformations and the lunes of Hippocrates.

Juliusz Brzeziński

Chapter 17. Proofs of the Theorems

Abstract

This chapter contains the proofs to all theorems presented in the book. Only a few theorems, which are typically covered in an introductory course on groups, rings and fields are proved in the Appendix. A proof of the fundamental theorem of algebra is given in connection with the exercises in Chap. 13.

Juliusz Brzeziński

Chapter 18. Hints and Answers

Abstract

This chapter contains hints and answers to all exercises presented in Chaps. 1–15 where an answer can be expected.

Juliusz Brzeziński

Chapter 19. Examples and Selected Solutions

Abstract

This chapters contains sample and in some cases complete solutions of some problems from Chaps. 1–15. We try to avoid solving standard problems formulated in these chapters, so that a solution is given as an example of how to handle similar problems to those presented in the text. Thus, the standard exercises in the text may be used as homework. Complete solutions to more special problems, as presented in this chapter, have at least three different functions. First, they contain a number of useful auxiliary results which are usually proved in the main texts of more standard textbooks. Second, some solutions are examples of how to work with the notions and approach similar problems (there is a rich selection of problems without solutions in Chap. 16). Third, some of the solutions presented in this chapter may be regarded as the last resort when serious attempts to solve a problem have been fruitless, or in order to compare one’s own solution to the one suggested in the book.

Juliusz Brzeziński

Backmatter

Titel: Galois Theory Through Exercises
verfasst von: Prof. Juliusz Brzeziński
Verlag: Springer International Publishing
Electronic ISBN: 978-3-319-72326-6
Print ISBN: 978-3-319-72325-9
DOI: https://doi.org/10.1007/978-3-319-72326-6