Galois Theory for Beginners: A Historical Perspective, Second Edition

242 Pages·2021·34.87 MB·English

by  Jörg Bewersdorff

STUDENT MATHEMATICAL LIBRARY
Volume 95
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Galois Theory
Galois theory is the culmination of a centuries-
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college mathematics, since the necessary ideas and properties of f
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groups and fi elds are provided as needed. Results in Galois theory r

are formulated fi rst in a concrete, elementary way, then in the B
modern form. Each chapter begins with a simple question that gives e
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the ancient problems of squaring the circle, duplicating the cube,
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and trisecting the angle, and the construction of regular n-gons are
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This new edition contains an additional chapter as well as twenty

facsimiles of milestones of classical algebra. It is suitable for E S
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STML/95
4-color cover: 238 pages spine: 1/2" fi nish size: 5.5” X 8.5” 50 lb stock
10.1090/stml/095
Galois Theory
for Beginners
A Historical Perspective
S e
econd dition

STUDENT MATHEMATICAL LIBRARY
Volume 95
Galois Theory
for Beginners
A Historical Perspective
S e
econd dition
Jörg Bewersdorff
Translated by David Kramer
EDITORIAL COMMITTEE
John McCleary Kavita Ramanan
Rosa C. Orellana John Stillwell (Chair)
2020 Mathematics Subject Classification. Primary 12-01; Secondary
01A05, 12-03, 12D05, 12F10.
First published in German under the title
Algebra fu¨r Einsteiger; Von der Gleichungsaufl¨osung zur Galois-Theorie
by J¨org Bewersdorff, edition: 6
(cid:2)c Springer Fachmedien Wiesbaden GmbH,
ein Teil von Springer Nature, 2019
This edition has been translated and published under license from
Springer Fachmedien Wiesbaden GmbH, part of Springer Nature.
Springer Fachmedien Wiesbaden GmbH, part of Springer Nature
takes no responsibility and shall not be made liable for the
accuracy of the translation.
Translated by David Kramer with additions
and corrections by the author.
Figure 1.1, p. 2, and Fugure 1.2, p. 4, are from Otto Neugebauer, Quellen
und Studien zur Geschichte der Mathematik, Astronomie und Physik,
Abteilung A: Quellen, Dritter Band, Dritter Teil. 1937, (cid:2)c Springer
Nature, reproduced with permission of SNCSC.
For additional information and updates on this book, visit
www.ams.org/bookpages/stml-95
Library of Congress Cataloging-in-Publication Data
Names: Bewersdorff,Jo¨rg, author.
Title: Galoistheoryforbeginners: ahistoricalperspective/J¨orgBewersdorff;trans-
latedbyDavidKramer.
Description: Providence, Rhode Island : American Mathematical Society, [2021] | Se-
ries: Student mathematical library, 1520-9121 ; Volume 95 | Includes index. |
First published in German under the title Algebra fu¨r Einsteiger; Von der Gle-
ichungsaufl¨osung zurGalois-TheoriebyJ¨orgBewersdorff,edition: 6in2019.
Identifiers: LCCN2021011513 |ISBN9781470465001 (paperback) |(ebook)
Subjects: LCSH:Galoistheory. |Polynomials. |AMS:Fieldtheoryandpolynomials–
Instructionalexposition(textbooks,tutorialpapers,etc.). |Historyandbiography
–Historyofmathematics and mathematicians–General histories,sourcebooks. |
Field theory and polynomials – Historical. | Field theory and polynomials – Real
and complex fields – Polynomials: factorization. | Field theory and polynomials –
Fieldextensions–Separable extensions,Galoistheory.
Classification: LCCQA214.B49132021|DDC512/.32–dc23
LCrecordavailableathttps://lccn.loc.gov/2021011513
Copyright page continues on page 219.
Contents
PrefacetotheEnglishEdition vii
PrefacestotheGermanEditions ix
Chapter1. CubicEquations 1
Chapter2. CasusIrreducibilis:TheBirthoftheComplexNumbers 27
Chapter3. BiquadraticEquations 41
Chapter4. EquationsofDegreenandTheirProperties 47
TheFundamentalTheoremofAlgebra: PlausibilityandProof 53
Chapter5. TheSearchforAdditionalSolutionFormulas 57
Permutations 62
TheFundamentalTheoremonSymmetricPolynomials 67
RuffiniandtheGeneralEquationofFifthDegree 71
Chapter6. EquationsThatCanBeReducedinDegree 77
TheDecompositionofIntegerPolynomials 79
Eisenstein’sIrreducibilityCriterion 82
Chapter7. TheConstructionofRegularPolygons 85
ConstructionswithStraightedgeandCompass 91
TheClassicalConstructionProblems 97
v
vi Contents
Chapter8. TheSolutionofEquationsoftheFifthDegree 105
TheTransformationsofTschirnhausandofBringandJerrard 112
Chapter9. TheGaloisGroupofanEquation 117
ComputingtheGaloisGroup 137
AQuickCourseinCalculatingwithPolynomials 142
Chapter10. AlgebraicStructuresandGaloisTheory 147
GroupsandFields 152
TheFundamentalTheoremofGaloisTheory: AnExample 164
TheUnsolvabilityoftheClassicalConstructionProblems 180
Chapter11. GaloisTheoryAccordingtoArtin 183
Epilogue 199
Index 211
Preface to the English
Edition
This book is a translation of the second edition of my German book
AlgebrafürEinsteiger:VonderGleichungsauflösungzurGalois-Theorie,
Vieweg, 2004. TheoriginalGermaneditionhasbeenexpandedbythe
additionofexercises. Thegoalofthebookisdescribedintheoriginal
preface. In a few words it can be sketched as follows: Galois theory
is presented in the most elementary way, following the historical evo-
lution. The main focus is always the classical application to algebraic
equationsandtheirsolutionsbyradicals.IamgratefultoDavidKramer,
whodidmorethantranslatethepresentbook,havingalsoofferedsev-
eralsuggestionsforimprovements.MythanksarealsodirectedtoUlrike
Schmickler-Hirzebruch, ofVieweg, whofirstproposedatranslationto
theAmericanMathematicalSociety,andtoEdwardDunne,oftheAMS,
formanagingthetranslation.
JörgBewersdorff
Translator’sNote
I wish to express my appreciation to Jörg Bewersdorff for his helpful
collaborationonthetranslationandtothefollowingindividualsatthe
AmericanMathematicalSociety:EdwardDunneforentrustingmewith
vii
viii PrefacetotheEnglishEdition
thisproject,BarbaraBeetonforherfriendlyandintelligentTEXnicalsup-
port,andArleneO’Seanforhercarefulcopyeditingofthetranslation.
DavidKramer