2011 Rosenberger Fine Carstensen

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De Gruyter GraduateCeline Carstensen Benjamin Fine Gerhard RosenbergerAbstract AlgebraApplications to Galois Theory, Algebraic Geometry and CryptographyDe GruyterMathematics Subject Classification 2010: Primary: 12-01, 13-01, 16-01, 20-01; Secondary: 01-01, 08-01, 11-01, 14-01, 94-01.This book is Volume 11 of the Sigma Series in Pure Mathematics, Heldermann Verlag.ISBN 978-3-11-025008-4 e-ISBN 978-3-11-025009-1Library of Congress Cataloging-in-Publication Data Carstensen, Celin

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De Gruyter GraduateCeline CarstensenBenjamin FineGerhard RosenbergerAbstract AlgebraApplications to Galois Theory,Algebraic Geometry and CryptographyDe GruyterMathematics Subject Classification 2010: Primary: 12-01, 13-01, 16-01, 20-01; Secondary: 01-01,08-01, 11-01, 14-01, 94-01.This book is Volume 11 of the Sigma Series in Pure Mathematics, Heldermann Verlag.ISBN 978-3-11-025008-4e-ISBN 978-3-11-025009-1Library of Congress Cataloging-in-Publication DataCarstensen, Celine.Abstract algebra : applications to Galois theory, algebraic geo-metry, and cryptography / by Celine Carstensen, Benjamin Fine,and Gerhard Rosenberger.p. cm. (Sigma series in pure mathematics ; 11)Includes bibliographical references and index.ISBN 978-3-11-025008-4 (alk. paper)1. Algebra, Abstract. 2. Galois theory. 3. Geometry, Algebraic.4. Crytography. I. Fine, Benjamin, 1948 II. Rosenberger, Ger-hard. III. Title.QA162.C375 20115151.02dc222010038153Bibliographic information published by the Deutsche NationalbibliothekThe Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.2011 Walter de Gruyter GmbH & Co. KG, Berlin/New YorkTypesetting: Da-TeX Gerd Blumenstein, Leipzig, www.da-tex.dePrinting and binding: AZ Druck und Datentechnik GmbH, Kempten Printed on acid-free paperPrinted in Germanywww.degruyter.comPrefaceTraditionally, mathematics has been separated into three main areas; algebra, analysisand geometry. Of course there is a great deal of overlap between these areas. Forexample, topology, which is geometric in nature, owes its origins and problems asmuch to analysis as to geometry. Further the basic techniques in studying topologyare predominantly algebraic. In general, algebraic methods and symbolism pervadeall of mathematics and it is essential for anyone learning any advanced mathematicsto be familiar with the concepts and methods in abstract algebra.This is an introductory text on abstract algebra. It grew out of courses given toadvanced undergraduates and beginning graduate students in the United States andto mathematics students and teachers in Germany. We assume that the students arefamiliar with Calculus and with some linear algebra, primarily matrix algebra and thebasic concepts of vector spaces, bases and dimensions. All other necessary materialis introduced and explained in the book. We assume however that the students havesome, but not a great deal, of mathematical sophistication. Our experience is that thematerial in this can be completed in a full years course. We presented the materialsequentially so that polynomials and eld extensions preceded an in depth look atgroup theory. We feel that a student who goes through the material in these notes willattain a solid background in abstract algebra and be able to move on to more advancedtopics.The centerpiece of these notes is the development of Galois theory and its importantapplications, especially the insolvability of the quintic. After introducing the basic al-gebraic structures, groups, rings and elds, we begin the theory of polynomials andpolynomial equations over elds. We then develop the main ideas of eld extensionsand adjoining elements to elds. After this we present the necessary material fromgroup theory needed to complete both the insolvability of the quintic and solvabilityby radicals in general. Hence the middle part of the book, Chapters 9 through 14 areconcerned with group theory including permutation groups, solvable groups, abeliangroups and group actions. Chapter 14 is somewhat off to the side of the main themeof the book. Here we give a brief introduction to free groups, group presentationsand combinatorial group theory. With the group theory material in hand we returnto Galois theory and study general normal and separable extensions and the funda-mental theorem of Galois theory. Using this we present several major applicationsof the theory including solvability by radicals and the insolvability of the quintic, thefundamental theorem of algebra, the construction of regular n-gons and the famousimpossibilities; squaring the circling, doubling the cube and trisecting an angle. Wevi Prefacenish in a slightly different direction giving an introduction to algebraic and groupbased cryptography.October 2010 Celine CarstensenBenjamin FineGerhard RosenbergerContentsPreface v1 Groups, Rings and Fields 11.1 Abstract Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Integral Domains and Fields . . . . . . . . . . . . . . . . . . . . . . 41.4 Subrings and Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Factor Rings and Ring Homomorphisms . . . . . . . . . . . . . . . . 91.6 Fields of Fractions . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.7 Characteristic and Prime Rings . . . . . . . . . . . . . . . . . . . . . 141.8 Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Maximal and Prime Ideals 212.1 Maximal and Prime Ideals . . . . . . . . . . . . . . . . . . . . . . . 212.2 Prime Ideals and Integral Domains . . . . . . . . . . . . . . . . . . . 222.3 Maximal Ideals and Fields . . . . . . . . . . . . . . . . . . . . . . . 242.4 The Existence of Maximal Ideals . . . . . . . . . . . . . . . . . . . . 252.5 Principal Ideals and Principal Ideal Domains . . . . . . . . . . . . . . 272.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Prime Elements and Unique Factorization Domains 293.1 The Fundamental Theorem of Arithmetic . . . . . . . . . . . . . . . 293.2 Prime Elements, Units and Irreducibles . . . . . . . . . . . . . . . . 353.3 Unique Factorization Domains . . . . . . . . . . . . . . . . . . . . . 383.4 Principal Ideal Domains and Unique Factorization . . . . . . . . . . . 413.5 Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.6 Overview of Integral Domains . . . . . . . . . . . . . . . . . . . . . 513.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514 Polynomials and Polynomial Rings 534.1 Polynomials and Polynomial Rings . . . . . . . . . . . . . . . . . . . 534.2 Polynomial Rings over Fields . . . . . . . . . . . . . . . . . . . . . . 554.3 Polynomial Rings over Integral Domains . . . . . . . . . . . . . . . . 574.4 Polynomial Rings over Unique Factorization Domains . . . . . . . . 584.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65viii Contents5 Field Extensions 665.1 Extension Fields and Finite Extensions . . . . . . . . . . . . . . . . . 665.2 Finite and Algebraic Extensions . . . . . . . . . . . . . . . . . . . . 695.3 Minimal Polynomials and Simple Extensions . . . . . . . . . . . . . 705.4 Algebraic Closures . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.5 Algebraic and Transcendental Numbers . . . . . . . . . . . . . . . . 755.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786 Field Extensions and Compass and Straightedge Constructions 806.1 Geometric Constructions . . . . . . . . . . . . . . . . . . . . . . . . 806.2 Constructible Numbers and Field Extensions . . . . . . . . . . . . . . 806.3 Four Classical Construction Problems . . . . . . . . . . . . . . . . . 836.3.1 Squaring the Circle . . . . . . . . . . . . . . . . . . . . . . . 836.3.2 The Doubling of the Cube . . . . . . . . . . . . . . . . . . . 836.3.3 The Trisection of an Angle . . . . . . . . . . . . . . . . . . . 836.3.4 Construction of a Regular n-Gon . . . . . . . . . . . . . . . . 846.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 897 Kroneckers Theorem and Algebraic Closures 917.1 Kron