Galois Theory for Beginners: A Historical Perspective

199 Pages·2006·11.939 MB·English

by  Jorg Bewersdorff

Galois
Theory
for
Beginners
A Historical
Perspective
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STUDENT MATHEMATICAL LIBRARY
Volume 35
Galois
Theory
for
Beginners
A Historical
Perspective
Bewersdorff
Jérg
Translated by David Kramer
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AMERICAN MATHEMATICAL SOCIETY
Providence, Rhoda Island
Editorial Board
Gerald B. Folland Brad Osgood
Robin Forman (Chair) Michael Starbird
Originally published in the German language by
Friedr. Vieweg 85 Sohn Verlag, 65189 Wiesbaden, Germany,
(2nd
“Jorg Bewersdorff: Algebra fiir Einsteiger. 2. Auflage edition)”.
as
@ Friedr. Vieweg & Sohn Verlag|GWV Fachverlage GmbH,
Wiesbaden, 2004
Translated by David Kramer with additions and corrections
by the author.
2000 Mathematics Subject ClassificationP.rimary 12-01;
Secondary 12F10.
For additional information and updates this book, visit
on
www.ams.org/bookpages/stml-35
Library of Congress Cataloging-in—Publication Data
Bewersdorff, Jorg.
Algebra fiir Einsteiger. English
Galois theory for beginners historical perspective / Jorg Bewersdorff
: a ;
translated by David Kramer.
(Student mathematical library, ISSN 1520-9121 35)
p. cm. -— ', V.
Includes index.
ISBN-13: 978—0—8218—3817-(4acid-free paper)
1. Galois theory. 2. Polynomials. I. Title.
QA214 .B49 2006
512’.32—dc22 2006048423
Copying and reprinting. Individual readers of this publication, and nonprofit
libraries acting for them, permitted to make fair of the material, such to
are use as
copy a chapter for use in teaching or research. Permission is granted to quote brief
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passages
the is given.
source
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Requests for such permission should be addressed to the Acquisitions Department,
American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-
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© 2006 by the American Mathematical Society. All rights reserved.
The American Mathematical Society retains all rights
except those granted to the United States Government.
Printed in the United States of America.
The used in this book is acid-free and falls within the guidelines
paper
established to and durability.
ensure permanence
Visit the AMS home at http: //www.ams.org/
page
10987654321 111009080706
Contents
Preface to the English Edition
vii
Prefaces to the German Editions
ix
Chapter 1. Cubic
Equations
Chapter 2. Casus Irreducibilis: The Birth of the Complex
Numbers
Chapter 3. Biquadratic Equations
23
Chapter 4. Equations of Degree and Their Properties 27
n
The Fundamental Theorem of Algebra: Plausibility
and Proof
32
Chapter 5. The Search for Additional Solution Formulas 37
Permutations
42
The Fundamental Theorem Symmetric
on
Polynomials
47
Ruffini and the General Equation of Fifth Degree 50
i
Contents
Vi
Chapter 6. Equations That Can Be Reduced in Degree 55
The Decomposition of Integer Polynomials 57
Eisenstein’s Irreducibility Criterion 60
Chapter 7. The Construction of Regular Polygons 63
Constructions with Straightedge and Compass 69
The Classical Construction Problems 74
Chapter 8. The Solution of Equations of the Fifth Degree 81
The Transformations of Tschirnhaus and of Bring and
J errard 89
Chapter 9. The Galois Group of Equation 93
an
Computing the Galois Group 114
A Quick Course in Calculating with Polynomials 119
Chapter 10. Algebraic Structures and Galois Theory 125
Groups and Fields 130
The Fundamental Theorem of Galois Theory: An
Example 144
Artin’s Version of the Fundamental Theorem of
Galois Theory 149
The Unsolvability of the Classical Construction
Problems 161
Epilogue
165
Index
177
Preface the
to English
Edition
This book is translation of the second edition of German book
a my
Algebra fiir EinsteigerV.‘on dew" GleichungsaufliisungGaloz's- Theo-
zur
rie, Vieweg, 2004. The original German edition has been expanded
by the addition of exercises. The goal of the book is described in
the original preface. In few words it be sketched follows:
a can as
Galois theory is presented in the most elementary following the
way,
historical evolution. The main focus is always the classical application
to algebraic equations and their solutions by radicals. I grateful
am
to David Kramer, who did than translate the present book,
more
having also offered several suggestions for improvements. My thanks
also directed to Ulrike Schmickler—HirzebruofchV,ieweg, who
are
first proposed translation to the American Mathematical Society,
a
and to Edward Dunne, of the AMS, for managing the translation.
Jorg Bewersdorff
Translator’s Note
I wish to appreciation to Jorg Bewersdorff for his helpful
express my
collaboration the translation and to the following individuals at
on
the American Mathematical Society: Edward Dunne for entrusting
vii
viii Preface to the English Edition
7
with this project, Barbara Beeton for her friendly and intelligent
me
and Arlene O’Sean for her careful copyediting of
’IEXnicaslupport,
the translation.
Kramer
‘David
E>
Pregfaces to the German
Editions
Math is like love; a simple idea, but it can get complicated.
R. Drabek
—
Preface to the First German Edition
The subject of this book is the history of classical problem in alge—
a
bra. We will recount the search for formulas describing the solutions
of polynomial equations in unknown and how succession of fail—
one a
ures led finallyto knowledge of a quite unexpected sort, and indeed,
of fundamental importance in mathematics.
Let look brieflyat the object that enticed of the world’s
us
many
best mathematicians period of three centuries. Perhaps, dear
over a
reader, recall from school days quadratic equations of the
you your
form
zn2—6:1c+1=O
well the “quadratifcormula”
as as
2
a;1,2=—3:-§i
for the solution of the “generalq”uadratic equation
2:2
+ + 0.
pm q =
IX
Prefaces to the German Editions
X
If apply this formula to example, obtain the two solu-
our we
we
1
tions
=3+2\/i :3~—2\/§.
and
$1 51:2
If interested in numerical solution, pull out
you are a you can
handy pocket calculator (or perhaps knowlhow to
your you com-
pute roots by hand) and obtain the decimal representations
square
5.828427. and 0.171572. You could also cal-
:01 = 332 = use your
. . . . .
culator to verify that these Values in fact solutions to the original
are
equation. A skeptic who wished to verify that the solutions derived
from the formula the exact solutions would have to substitute
are
the expressions containing the roots into the equation and
square
demonstrate that the quadratic polynomial 11:2 6:1: + 1 0 actually
— 2
vanishes—that is, the value zero—at the values and
assumes :1: = $1
33 = (I32.
The Solution of Equations of Higher Degree. It has long been
known how to solve cubic equations such
as
:1:3—3x2—3:1:—1=0
by of formula similar to the quadratic formula. Indeed,
means a
such formulas first published in 1545 by Cardano (1501-1676)
were
in his book Ars Magna. However, they quite complicated, and
are
have little use for numerical calculation. In an age of practically
unlimited computing do without such explicit formulas
power, we can
in practical applications, since it sufiices completely to determine the
solutions by of numeric algorithms. Indeed, for such
means
every
variable
equation in single there exist approximation methods that
a
iteratively, that is, step by step, compute the desired solution
more
and precisely. Such procedure is until the solution has
more a run
reached an accuracy suitable for the given application.
However such iterative numeric procedures unsuitable when
are
not only the numerical Value of solution is sought, such
a as 931 =
3.847322. in the previous example, but the “exact” value
. .
$131+?/§+\3/Z.
It is not only that such algebraic representation certain
an possesses a
aesthetic quality, but in addition, numeric solution is insufficient if
a
Preface to the First German Edition xi
0
hopes to derive mathematical knowledge and principles from the
one
solution of the equation. Let hypothesize, for example, based
us on
numeric calculation, the following identities:
6/w_1=§(enva+
V13),
163
e"
262537412640768744,
=
and
27r 1 1 1
\/7
2 : 8\/17 : 3 4 2 1
cos =
17 8 8
+%\/17+3\/fi—\/34—\/?7—2\/34+2\/E.
Without going into detail, it plausible that behind such
seems
identities, if indeed they correct, lie mathematical laws.
are some
A direct check to determine whether they in fact correct
are or are
merely the result of chance numeric approximation would be difficult.1
But back to Cardano. In addition to the solution for cubic
equa-
tions, Cardano published in his Ars Magna general formula for
a
quartic equations, that is, equations of the fourth degree, also known
biquadratic equations. Using such formulas, the equation
as
a:4—8m+6=0
11
will reveal that only the first and third identities correct. The first
are was
discovered by the Indian mathematician Ramanujan (1887-1920) and be easily
can
checked. The third, which will be discussed in Chapter 7, contains within it proof that
a
the regular heptadecagon (seventeen—sipdoelydgon) be constructed with straight-
can
edge and
compass.
The second equation is not exact. The actual value of the right—handside is
262537412640768743.9999999999992501
. . . .
However, this approximate identity is than chance. It is based
more mere on some
deep number—theoretic relationships. For this, Philip J. Davies, Are there
more on see
coincidences in mathematics? American Mathematical Monthly 88 (1981), 311~
pp.
320.