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STUDENT MATHEMATICAL LIBRARY Volume 99 Finite Fields, with Applications to Combinatorics Kannan Soundararajan Finite Fields, with Applications to Combinatorics STUDENT MATHEMATICAL LIBRARY Volume 99 Finite Fields, with Applications to Combinatorics Kannan Soundararajan EDITORIAL COMMITTEE John McCleary Paul Pollack Rosa C. Orellana (Chair) Kavita Ramanan 2020 Mathematics Subject Classiﬁcation. Primary 11-01, 05-01, 12-01, 11A07, 11A51, 05B10, 12E20. For additional information and updates on this book, visit www.ams.org/bookpages/stml-99 Library of Congress Cataloging-in-Publication Data Cataloging-in-Publication Data has been applied for by the AMS. See http://www.loc.gov/publish/cip/. DOI: https://doi.org/10.1090/stml/99 Copying and reprinting. Individual readers of this publication, and nonproﬁt li- braries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passagesfromthispublicationinreviews,provided thecustomaryacknowledgment of thesourceisgiven. Republication,systematiccopying,ormultiplereproductionofanymaterialinthis publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/ publications/pubpermissions. Send requests for translation rights and licensed reprints to reprint-permission@ ams.org. (cid:2)c 2022 by the author. All rights reserved. Printed in the United States of America. (cid:2)∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 ToWaheedaandKesi Contents Preface xi Chapter1. Primesandfactorization 1 §1.1. Groups 1 §1.2. Rings 4 §1.3. Integraldomainsandfields 6 §1.4. Divisibility: primesandirreducibles 9 §1.5. IdealsandPrincipalIdealDomains(PIDs) 12 §1.6. Greatestcommondivisors 13 §1.7. Uniquefactorization 15 §1.8. Euclideandomains 17 §1.9. Exercises 21 Chapter2. Primesintheintegers 27 §2.1. Theinfinitudeofprimes 27 §2.2. Bertrand’spostulate 32 §2.3. Howmanyprimesarethere? 38 §2.4. Exercises 41 Chapter3. Congruencesinrings 45 §3.1. Congruencesandquotientrings 45 vii viii Contents §3.2. Theringℤ/𝑛ℤ 49 §3.3. Primeidealsandmaximalideals 51 §3.4. PrimesintheGaussianintegers 55 §3.5. Exercises 58 Chapter4. Primesinpolynomialrings: constructingfinitefields 63 §4.1. Primesinthepolynomialringoverafield 63 §4.2. AnanalogueoftheproofofBertrand’spostulate 68 §4.3. AnanalogueofEuler’sproof 71 §4.4. Möbiusinversionandaformulafor𝜋(𝑛;𝔽 ) 74 𝑞 §4.5. Exercises 79 Chapter5. Theadditiveandmultiplicativestructuresoffinite fields 83 §5.1. Moreaboutgroups: cyclicgroups 83 §5.2. Moreaboutgroups: Lagrange’stheorem 87 §5.3. Theadditivestructureoffinitefields 90 §5.4. Themultiplicativestructureoffinitefields 95 §5.5. Exercises 97 Chapter6. Understandingthestructureofℤ/𝑛ℤ 99 §6.1. TheChineseRemainderTheorem 99 §6.2. Thestructureofthemultiplicativegroup(ℤ/𝑛ℤ)× 103 §6.3. Existenceofprimitiveroots mod𝑝𝑒: ProofofTheorem 6.10 105 §6.4. Exercises 108 Chapter7. Combinatorialapplicationsoffinitefields 111 §7.1. Sidonsetsandperfectdifferencesets 111 §7.2. ProofofTheorem7.3 116 §7.3. TheErdős-Turánbound—ProofofTheorem7.4 117 §7.4. Perfectdifferencesets—ProofofTheorem7.8 121 §7.5. Alittlemoreonfinitefields 124 §7.6. DeBruijnsequences 126 §7.7. Amagictrick 129