Algebra: Polynomials, Galois Theory and Applications

328 Pages·2017·17.42 MB·English

by  Frédéric Butin

Algebra
Polynomials, Galois Theory and Applications
Algebra
Polynomials, Galois Theory and Applications
Frédéric Butin
DOVER PUBLICATIONS, INC.
Mineola, New York
Copyright Copyright © 2017 by Frédéric Butin All rights reserved.
Bibliographical Note
This Dover edition, first published in 2017, is a new English translation of Algèbre — Polynômes, théorie
de Galois et applications informatiques, published in Paris by Hermann Editeurs in 2011.
Library of Congress Cataloging-in-Publication Data Names: Butin, Frédéric.
Title: Algebra : polynomials, Galois theory and applications / FrédéricButin.
Other titles: Algèbre. English Description: Dover edition. | Mineola, New York : Dover Publications, [2017]
| Series: Aurora Dover modern math originals | English translation of: Algèbre : Polynômes, théorie de
Galois et applications informatiques (Paris : Hermann Editeurs, 2011). Includes bibliographical
references and index.
Identifiers: LCCN 2016043751| ISBN 9780486810157 | ISBN 0486810151
Subjects: LCSH: Galois theory. | Polynomials. | Algebra, Abstract.
Classification: LCC QA214 .B8813 2017 | DDC 512/.32—dc23
LC record available at https://lccn.loc.gov/2016043751
Manufactured in the United States by LSC Communications 81015101  2017
www.doverpublications.com
Preface
Many students are familiar with Galois theory because they have learned that
equations of degree greater than or equal to five are not solvable by radicals, or
because they have heard about insolvable problems such as the Delian problem,
the angle trisection or the quadrature of the circle.
Luckily, Galois theory is not restricted to negative solutions of historical
problems! The theory makes many issues easier to understand through the
agreement that it establishes between the subfields of a given field and the
subgroups of the group of its automorphisms. It is also linked to practical
applications such as error-correcting codes that are widely used in digital storage
media.
This book is addressed to undergraduate and graduate students, to students
who are preparing for a master’s degree, and to anyone who wants to discover
the properties of this theory. Engineering students will also find a presentation of
the mathematical tools underlying the techniques that they use.
The aim of the book is to provide the reader with a clear and precise way to
use all the tools needed for their progress (the book is self-contained). The
proofs of the theorems and the exercises (they are all entirely solved) are
detailed to facilitate understanding. The book is divided into three main parts, as
well as a small number of chapters (10), which presents the topics with unity.
The first part introduces the essential notions of Galois theory. It reviews
arithmetic, cryptography, and the symmetric group in the first chapter; in the
second chapter, rings and polynomials, Euclidean division and the extended
algorithm are discussed, as well as multivariate, symmetric, and general
polynomials that are irreducible.
The second part is devoted to algebraic, normal and separable extensions,
Galois extensions (which are at the center of the theory), followed by abelian,
cyclic and radical extensions, as well as cyclotomic polynomials. After
completing the second part, the reader will have a good understanding of the
Galois group of a polynomial.
Applications of Galois theory are presented in the third part. Ruler and
compass constructions are followed by the study of irreducible polynomials over
finite fields, their factorization and applications—particularly in computer
science, where the error-correcting code of CDs is explicitly worked out—and
results about algebraic integers.
From the beginning of the book, the study takes place in a general framework
to provide the reader with a global view of the subject. The goal is not to engage
in abstract applications for fun, but to avoid masking the essential point by many
supplementary hypotheses due to particular cases. Starting from this general
framework, the book always aims for practical applications: therefore, the
calculations are entirely worked out, because if the theory seems transparent, this
is often due to insight gained by working through the relevant calculations.
Throughout the book we also make use of formal computation: many
applications suggest a different and wider view of problems and use a current
tool to provide examples that are too tedious to study “by hand.” The chosen
software is Maple, but other software programs can be used.
The book gives several results rarely found in the literature, such as the
Chebotarev theorem, the explicit study of error-correcting codes and the irre-
ducibility of the permanent.
A biography of quoted mathematicians is located at the end of the book,
which enables the reader to examine their mathematical culture.
I am grateful to Gadi S. Perets who helped me greatly with proofreading the
English text.
Frédéric Butin
Paris, France
Introduction
Solutions of certain polynomial equations can be written with square roots,
cubic roots, fourth roots ... (cf. solutions of equations of degree 2 with the
discriminant formula). However, it is well known that equations of degree
greater than or equal to 5 cannot, in general, be solved with this type of formula.
What is the reason?
Let us consider three examples. First, let us take the equation x2 – 2 = 0. Its two
roots in ℂ are and – , thus the smallest subfield of ℂ containing ℚ and the
roots of this equation is ℚ[ ], i.e., the set of numbers of the form a + b with
(a, b) ∈ ℚ2 (in fact we have ( ) ∈ ℚ and ). This is a vector space
over ℚ, and a basis of this space is (1, ).
Let f be an automorphism of the field ℚ[ ] that fixes the elements of ℚ. Then
we have
and we get two maps f and f determined by the relations f ( ) = and f ( )
1 2 1 2
= – . Conversely, these two maps are automorphisms of ℚ[ ]. Then, the
group of automorphisms of ℚ[ ] that fix the elements of ℚ is
This group is isomorphic to the symmetric group and, in particular, this is a
2
solvable group.
Now, let us consider the equation x4 – 7x2 + 10 = 0. Its roots are ± and ± .
The smallest subfield K of ℂ containing ℚ and the roots of this equation is the
set of numbers of the form a + b + c + d with (a, b, c, d) ∈ ℚ4. This is a
vector space over ℚ and a basis of this space is (1, , , ).
Let f be an automorphism of the field K that fixes the elements of ℚ. Then f
satisfies f( ) = ± and f( ) = ± . The group of automorphisms of K that fix
the elements of ℚ is thus {f , f , f , f }, with
1 2 3 4
This group is isomorphic to the group (ℤ 2ℤ) × (ℤ 2ℤ) and is solvable.
Let us finally study the equation x5 + 4x4 + x2 – 7 = 0, which possesses five
complex roots. Let K be the smallest subfield of ℂ containing ℚ and the roots of
this equation. We can show that the group of automorphisms of K that fix the
elements of ℚ is isomorphic to the symmetric group . Moreover, it turns out
5
that the roots of this equation cannot be written with square roots, cubic roots,
etc.
What can be deduced from these examples? Solutions of the first two equations
can be explicitly expressed with radicals: one says that these equations are
solvable by radicals. And the automorphism groups constructed for these
examples are solvable, which is not the case for . This is not a coincidence,
5
but a fundamental result of Galois theory that asserts that an equation is solvable
by radicals if and only if the automorphism group associated with it is solvable.
This automorphism group is called the Galois group of the equation.
As a preliminary to the study of such equations, we will introduce the
vocabulary related to algebraic extensions. An algebraic extension of a field K is
a field L whose K is a subfield and whose every element is a root of a
polynomial with coefficients in K. For example, the set of algebraic numbers is
an algebraic extension of ℚ. We will say that L has a finite degree over K if L is
a K-vector space of finite dimension.
Then we will focus our attention on normal extensions. One says that an
extension L of a field K is normal if every irreducible polynomial P ∈ K[X] that
possesses a root in L possesses all its roots in L. The extension ℚ[ ] introduced
for studying the first equation is an example of normal extension.
We will also study the problem of multiple roots, and for this purpose we will
introduce another type of algebraic extension, namely separable extensions. An
extension L of K is separable if every element of L is cancelled by a polynomial
P G K[X] that is coprime to its derivative. This will lead us to study finite fields.
as well as perfect fields.
Finally, we will have all the tools for studying Galois extensions, that is to say
extensions that are at the same time normal and separable. We will state the
fundamental theorem of Galois theory: if if is a field, L a Galois extension of
finite degree of K, and G the group of automorphisms of L that fix the elements
of K, then there exists a one-to-one correspondence between the subgroups of G
and the subextensions of L containing K. The automorphism group is called the
Galois group of L over K. This theorem is in a sense the heart of Galois theory,
in that it reduces the study of subfields of L to the study of subgroups of G.
After introducing new types of algebraic extensions, such as abelian, cyclic,
cyclotomic or radical extensions, we will prove another fundamental theorem,
namely the theorem quoted at the beginning of the introduction: an equation is
solvable by radicals if and only if its Galois group is solvable.
Armed with these tools, we will be able to show how Galois theory is useful,
providing applications in extremely diverse fields.
A first application is an answer to a question that one can naturally ask: when is
a given group the Galois group of an equation? We will answer in a (very)
partial way, knowing that the general problem is still open.
Galois theory answers another algebraic problem, which consists of searching
integers in number fields. One says that a complex number is an algebraic
integer if it is a root of a monic polynomial with integer coefficients. To know
which algebraic integers are contained in a given subfield of ℂ, we will
introduce two technical tools: the norm and the trace. Moreover, these tools will
enable us to obtain normal bases of Galois extensions L of finite degree of a field
K, i.e., bases of L whose elements form an orbit under the action of the Galois
group of L over K.
Another well-known application of this theory is the negative answer that it
provides to historical questions of constructions with a ruler and a compass, such
as the Delian problem, the angle trisection and the quadrature of the circle.
Through the Galois correspondence, we will also know which regular polygons
can be constructed with a ruler and a compass.
In contrast to these questions that already intrigued the ancient world, the Galois
theory is of utmost interest in the current problem of error-correcting codes:
during the transmission of a message, errors often appear because the
communication channel is not totally reliable. An error-correcting code is a
computer tool that enables us to code the message before sending it, and also to
recover the original message after having corrected the transmission errors. The
framework of finite fields—whose classification results from Galois theory—is
the foundation of the study of these codes. Among the many concrete areas
where these codes are used, the case of CDs, and more generally, any digital
storage media, will be studied in detail.