Galois Groups and Field Extensions for Solvable Quintics E-Book

Achim Plum

• Herausgeber: Books on Demand
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 52

Veröffentlichungsjahr: 2022

Beschreibung

In the context of self-studies on polynomials of degree 5 (quintics), Galois groups and field extensions related to solvable quintics - irreducible and reducible ones - have been analyzed. Effort has been spent to provide comprehensive explanations and some examples of those structures; the intention is to give the reader an overview and a decent insight into the topic, which may help to understand the algebraic structures and their arithmetics better.

Leseprobe

Galois Groups and Field Extensions for Solvable Quintics

Galois Groups and Field Extensions for Solvable Quintics1   Introduction2   Elements of Galois Theory3   Irreducible Quintics4   Reducible Quintics5   Summary6   ReferencesCopyright

Galois Groups and Field Extensions for Solvable Quintics

Abstract

In the context of self-studies on polynomials of degree 5 (quintics), Galois groups and field extensions related to solvable quintics - irreducible and reducible ones - have been analyzed. Effort has been spent to provide comprehensive explanations and some examples of those structures; the intention is to give the reader an overview and a decent insight into the topic, which may help to understand the algebraic structures and their arithmetics better.

Quite some reference materials of relevant or related topics are listed at the end of the e-book, primarily referring to Wikipedia articles. This ebook was written according to the best knowledge of the author. To foster further learning and exchange on the topic and the mathematical content, the author is interested in feedback for improvements or extensions and can be contacted at the email address [email protected].

1   Introduction

This e-book focuses on solvable irreducible and reducible quintics - polynomials of degree 5 - and how Galois groups can be determined. Galois theory laid the foundation for understanding whether polynomials can be solved by radicals or not: if the Galois group is solvable, the polynomial is as well. Nevertheless, the determination of Galois groups and corresponding field extensions is not always easy.

The following second chapter provides a compact overview of the theory of Galois groups and field extensions. See [2,3,4] for some background on group theory and finite groups, and [1] for an overview of Galois theory. It is key to understand the concept of field automorphisms and the role the field ℚ plays in this context; actually, the base field normally chosen is ℚ, which implies that all irrational zeroes generate field extensions of ℚ, and many polynomials are irreducible over ℚ [10]. The concepts are explained in general as well as with some examples.

When analyzing a polynomial, the splitting field [14] is of highest interest. It is constructed by a chain of field extensions created by irreducible factors of the polynomial over a given base field F. Conceptually, there is an isomorphism between the quotient ring of the commmutative ring of polynomials over the maximal ideal [8] generated by an irreducible polynomial to the field created by adjoining a zero of that polynomial (i.e. adding with closure under addition and multiplication [12]). The convention to write this down for an irreducible polynomial f(x) over a field F is F[x]/(f(x)), with F[x] being the ring of polynomials with coefficients in F and (f(x)) denoting the ideal generated by f(x). Such a quotient ring is a field that is isomorphic to F(n), with n being a zero of f(x), which is the field generated by adjoining this number to the base field F. The splitting field is created by repeating this process with sufficiently many additional numbers (e.g. zeroes of f(x) or primitive roots of unity) such that all zeroes of f(x) are elements of the resulting field. In this e-book, the way to denote the splitting field of f(x) is SpF(f(x)).

In chapter 3, several classes of irreducible quintics are considered, and the Galois groups and field extensions for those are derived. Irreducible quintics are the ones that cannot be split into factors of smaller degree with coefficients of the base field ℚ and thus are of high interest in Galois theory. The roots of unity play a key role in how field automorphisms can actually operate, in particular on the zeroes. Galois groups of polynomials are sometimes denoted as Gal(f(x)) here, which is equivalent to (and a short form of) Gal(SpF(f(x))).

Chapter 4 covers reducible quintics. The fact that these can be split into polynomials of lower degree does not make them less interesting; it is not very easy to identify such quintics (see e.g. [21,22] for formulas), and the Galois groups and respective field extensions can have a variety of different structures.

This e-book is offered to readers who are advancing in mathematical studies, especially in topics of algebra, and who are interested in the link between solvable quintics and Galois theory. As a prerequisite, the reader should have some background  on group and field theory [2] and a solid understanding of the field of complex numbers [15]. A list of references at the end provides a useful source of further reading materials.

2   Elements of Galois Theory

In Wikipedia, some comprehensive texts can be found that explain Galois theory [1]; they include references to relevant theory books as well. Some of the key concepts are explained here for convenient reading. In order to get deeper insights into the theory and its applications, there is a list of references for further reading at the end.

Field Automorphisms

A key concept of Galois theory is a correspondence between field extensions and groups in group theory. It makes a lot of sense to start with the base field ℚ of rational numbers; irrational zeroes in ℝ\ℚ or any zeroes in ℂ\ℝ can be treated uniformly. The Galois group of a polynomial f(x) ∈ ℚ[x] corresponds to the splitting field F/ℚ (F over ℚ) of the polynomial, which means that the polynomial splits into linear factors in F/ℚ, and the splitting field F/ℚ is the smallest and unique field extension of ℚ with that property. By this, the Galois group of f(x) is defined as the Galois group of F/ℚ. [14]

According to [1], the Galois group of a field F is the group of field automorphisms of F that keep ℚ fixed. A field automorphism φ has two crucial properties:

It is an endomorphism (a homomorphism of an algebraic structure to itself [9]) operating on the field extension F=F/ℚ as a vector space, which is created by adjoining numbers [12,13] that are not in ℚ. As such, it has the well-known feature of vector space homomorphisms that it is linear in terms of addition in that φ(x+y)=φ(x)+φ(y) and φ(q∙x)=q∙φ(x) for all elements x,y ∈ F and q ∈ ℚ.

Being a field homomorphism adds another strong feature because it ensures that multiplication can be  executed likewise before or after the automorphism, with the same result: φ(x∙y)=φ(x)∙φ(y) for all elements x,y ∈ F.

Field automorphisms in ℂ/ℝ

Statement: For ℂ/ℝ, the identity function and complex conjugation are the only field automorphisms.

Proof: It is trivial that id