sign in sign up
<<
Home , Euclidean algorithm

Advanced Modern Algebra

Third Edition, Part 1

Joseph J. Rotman

Graduate Studies in Volume 165

American Mathematical Society Advanced Modern Algebra Third Edition, Part 1

https://doi.org/10.1090//gsm/165

Advanced Modern Algebra Third Edition, Part 1

Joseph J. Rotman

Graduate Studies in Mathematics Volume 165

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Dan Abramovich Daniel S. Freed Rafe Mazzeo (Chair) Gigliola Staffilani

The 2002 edition of this book was previously published by Pearson Education, Inc.

2010 Mathematics Subject Classification. Primary 12-01, 13-01, 14-01, 15-01, 16-01, 18-01, 20-01.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-165

Library of Congress Cataloging-in-Publication Rotman, Joseph J., 1934– Advanced modern algebra / Joseph J. Rotman. – Third edition. volumes cm. – (Graduate studies in mathematics ; volume 165) Includes bibliographical references and index. ISBN 978-1-4704-1554-9 (alk. paper : pt. 1) 1. Algebra. I. Title. QA154.3.R68 2015 512–dc23 2015019659

Copying and reprinting. Individual readers of this publication, and nonprofit libraries 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 passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. Third edition c 2015 by the American Mathematical Society. All rights reserved. Second edition c 2010 by the American Mathematical Society. All rights reserved. First edition c 2002 by the American Mathematical Society. All right reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ 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 http://www.ams.org/ 10987654321 201918171615 To my wife Marganit and our two wonderful kids Danny and Ella, whom I love very much

Contents

Preface to Third Edition: Part 1 xi Acknowledgments xiv

Part A. Course I

Chapter A-1. Classical Formulas 3 Cubics 4 Quartics 6

Chapter A-2. Classical Theory 9 Divisibility 9 Euclidean 16 Congruence 19

Chapter A-3. Commutative Rings 29 41 Homomorphisms 47 Rings 55 From Arithmetic to Polynomials 62 Maximal Ideals and Prime Ideals 74 Finite Fields 83 Irreducibility 89 Euclidean Rings and Principal Domains 97 Unique Domains 104

Chapter A-4. Groups 115 Permutations 116

vii viii Contents

Even and Odd 123 Groups 127 Lagrange’s Theorem 139 Homomorphisms 150 Quotient Groups 159 Simple Groups 173

Chapter A-5. Galois Theory 179 Insolvability of the Quintic 179 Classical Formulas and Solvability by Radicals 187 Translation into Theory 190 Fundamental Theorem of Galois Theory 200 Calculations of Galois Groups 223

Chapter A-6. Appendix: Set Theory 235 Equivalence Relations 243

Chapter A-7. Appendix: Linear Algebra 247 Vector Spaces 247 Linear Transformations and Matrices 259

Part B. Course II

Chapter B-1. Modules 273 Noncommutative Rings 273 Chain Conditions on Rings 282 Left and Right Modules 288 Chain Conditions on Modules 300 Exact Sequences 305

Chapter B-2. Zorn’s Lemma 313 Zorn, Choice, and Well-Ordering 313 Zorn and Linear Algebra 319 Zorn and Free Abelian Groups 323 Semisimple Modules and Rings 334 Algebraic Closure 339 Transcendence 345 L¨uroth’s Theorem 353

Chapter B-3. Advanced Linear Algebra 359 Torsion and Torsion-free 359 Basis Theorem 362 Contents ix

Fundamental Theorem 371 Elementary 371 Invariant Factors 374 From Abelian Groups to Modules 378 Rational Canonical Forms 383 Eigenvalues 388 Jordan Canonical Forms 395 Smith Normal Forms 402 Inner Product Spaces 417 Orthogonal and Symplectic Groups 429 Hermitian Forms and Unitary Groups 436

Chapter B-4. Categories of Modules 441 Categories 441 Functors 461 Galois Theory for Infinite Extensions 475 Free and Projective Modules 481 Injective Modules 492 Divisible Abelian Groups 501 Tensor Products 509 Adjoint Isomorphisms 522 Flat Modules 529

Chapter B-5. Multilinear Algebra 543 Algebras and Graded Algebras 543 Tensor Algebra 552 Exterior Algebra 561 Grassmann Algebras 566 Exterior Algebra and Differential Forms 573 575

Chapter B-6. Commutative Algebra II 591 Old-Fashioned Algebraic Geometry 591 Affine Varieties and Ideals 593 Nullstellensatz 599 Nullstellensatz Redux 604 Irreducible Varieties 614 Affine Morphisms 623

Algorithms in k[x1,...,xn] 628 Monomial Orders 629 x Contents

Division 636 Gr¨obner Bases 639

Chapter B-7. Appendix: Categorical Limits 651 Inverse Limits 651 Direct Limits 657 Directed Index Sets 659 Adjoint Functors 666

Chapter B-8. Appendix: Topological Spaces 673 Topological Groups 678

Bibliography 681

Special Notation 687

Index 693 Preface to Third Edition: Part 1

Algebra is used by virtually all , be they analysts, combinatorists, computer scientists, geometers, logicians, number theorists, or topologists. Nowa- days, everyone agrees that some knowledge of linear algebra, group theory, and commutative algebra is necessary, and these topics are introduced in undergrad- uate courses. Since there are many versions of undergraduate algebra courses, I will often review definitions, examples, and theorems, sometimes sketching proofs and sometimes giving more details.1 Part 1 of this third edition can be used as a text for the first year of graduate algebra, but it is much more than that. It and the forthcoming Part 2 can also serve more advanced graduate students wishing to learn topics on their own. While not reaching the frontiers, the books provide a sense of the successes and methods arising in an area. In addition, they comprise a reference containing many of the standard theorems and definitions that users of algebra need to know. Thus, these books are not merely an appetizer, they are a hearty meal as well. When I was a student, Birkhoff–Mac Lane, A Survey of Modern Algebra [8], was the text for my first algebra course, and van der Waerden, Modern Algebra [118], was the text for my second course. Both are excellent books (I have called this book Advanced Modern Algebra in homage to them), but times have changed since their first publication: Birkhoff and Mac Lane’s book appeared in 1941; van der Waerden’s book appeared in 1930. There are today major directions that either did not exist 75 years ago, or were not then recognized as being so important, or were not so well developed. These new areas involve algebraic geometry, category

1It is most convenient for me, when reviewing earlier material, to refer to my own text FCAA: A First Course in ,3rded.[94], as well as to LMA, the book of A. Cuoco and myself [23], Learning Modern Algebra from Early Attempts to Prove Fermat’s Last Theorem.

xi xii Preface to Third Edition: Part 1

theory,2 computer science, homological algebra, and representation theory. Each generation should survey algebra to make it serve the present time. The passage from the second edition to this one involves some significant changes, the major change being organizational. This can be seen at once, for the elephantine 1000 page edition is now divided into two volumes. This change is not merely a result of the previous book being too large; instead, it reflects the structure of beginning graduate level algebra courses at the University of Illinois at Urbana–Champaign. This first volume consists of two basic courses: Course I (Galois theory) followed by Course II (module theory). These two courses serve as joint prerequisites for the forthcoming Part 2, which will present more advanced topics in theory, group theory, algebraic , homological algebra, representation theory, and algebraic geometry. In addition to the change in format, I have also rewritten much of the text. For example, noncommutative rings are treated earlier. Also, the section on alge- braic geometry introduces regular functions and rational functions. Two proofs of the Nullstellensatz (which describes the maximal ideals in k[x1,...,xn]whenk is an algebraically closed field) are given. The first proof, for k = C (which easily generalizes to uncountable k), is the same proof as in the previous edition. But the second proof I had written, which applies to countable algebraically closed fields as well, was my version of Kaplansky’s account [55] of proofs of Goldman and of Krull. I should have known better! Kaplansky was a master of exposition, and this edition follows his proof more closely. The reader should look at Kaplansky’s book, Selected Papers and Writings [58], to see wonderful mathematics beautifully expounded. I have given up my attempted spelling reform, and I now denote the ring of mod m by Zm instead of by Im. A star * before an exercise indicates that it will be cited elsewhere in the book, possibly in a proof. The first part of this volume is called Course I; it follows a syllabus for an actual course of lectures. If I were king, this course would be a transcript of my lectures. But I am not king and, while users of this text may agree with my global organization, they may not agree with my local choices. Hence, there is too much material in the Galois theory course (and also in the module theory course), because there are many different ways an instructor may choose to present this material. Having lured students into beautiful algebra, we present Course II: module theory; it not only answers some interesting questions (canonical forms of matrices, for example) but it also introduces important tools. The content of a sequel algebra course is not as standard as that for Galois theory. As a consequence, there is much more material here than in Course I, for there are many more reasonable choices of material to be presented in class. To facilitate various choices, I have tried to make the text clear enough so that students can read many sections independently. Here is a more detailed description of the two courses making up this volume.

2A Survey of Modern Algebra was rewritten in 1967, introducing categories, as Mac Lane– Birkhoff, Algebra [73]. Preface to Third Edition: Part 1 xiii

Course I

After presenting the cubic and quartic formulas, we review some undergraduate number theory: algorithm; Euclidian algorithms (finding d =gcd(a, b) and expressing it as a ), and congruences. Chapter 3 begins with a review of commutative rings, but continues with maximal and prime ideals, finite fields, irreducibility criteria, and euclidean rings, PIDs, and UFD’s. The next chapter, on groups, also begins with a review, but it continues with quotient groups and simple groups. Chapter 5 treats Galois theory. After introducing Galois groups of extension fields, we discuss solvability, proving the Jordan-H¨older Theorem and the Schreier Refinement Theorem, and we show that the general quintic is not solvable by radicals. The Fundamental Theorem of Galois Theory is proved, and applications of it are given; in particular, we prove the Fundamental Theorem of Algebra (C is algebraically closed). The chapter ends with computations of Galois groups of polynomials of small degree. There are also two appendices: one on set theory and equivalence relations; the other on linear algebra, reviewing vector spaces, linear transformations, and matrices.

Course II

As I said earlier, there is no commonly accepted syllabus for a sequel course, and the text itself is a syllabus that is impossible to cover in one semester. However, much of what is here is standard, and I hope instructors can design a course from it that they think includes the most important topics needed for further study. Of course, students (and others) can also read chapters independently. Chapter 1 (more precisely, Chapter B-1, for the chapters in Course I are labeled A-1, A-2, etc.) introduces modules over noncommutative rings. Chain conditions are treated, both for rings and for modules; in particular, the Hilbert Basis The- orem is proved. Also, exact sequences and commutative diagrams are discussed. Chapter 2 covers Zorn’s Lemma and many applications of it: maximal ideals; bases of vector spaces; subgroups of free abelian groups; semisimple modules; existence and uniqueness of algebraic closures; transcendence degree (along with a proof of L¨uroth’s Theorem). The next chapter applies modules to linear algebra, proving the Fundamental Theorem of Finite Abelian Groups as well as discussing canonical forms for matrices (including the Smith normal form which enables computation of invariant factors and elementary divisors). Since we are investigating linear al- gebra, this chapter continues with bilinear forms and inner product spaces, along with the appropriate transformation groups: orthogonal, symplectic, and unitary. Chapter 4 introduces categories and functors, concentrating on module categories. We study projective and injective modules (paying attention to projective abelian groups, namely free abelian groups, and injective abelian groups, namely divisible abelian groups), tensor products of modules, adjoint isomorphisms, and flat mod- ules (paying attention to flat abelian groups, namely torsion-free abelian groups). Chapter 5 discusses multilinear algebra, including algebras and graded algebras, tensor algebra, exterior algebra, Grassmann algebra, and determinants. The last xiv Preface to Third Edition: Part 1

chapter, Commutative Algebra II, has two main parts. The first part discusses “old-fashioned algebraic geometry,” describing the relation between zero sets of polynomials (of several variables) and ideals (in contrast to modern algebraic ge- ometry, which extends this discussion using sheaves and schemes). We prove the Nullstellensatz (twice!), and introduce the category of affine varieties. The second part discusses algorithms arising from the for polynomials of several variables, and this leads to Gr¨obner bases of ideals. There are again two appendices. One discusses categorical limits (inverse limits and direct limits), again concentrating on these constructions for modules. We also mention adjoint functors. The second appendix gives the elements of topological groups. These appendices are used earlier, in Chapter B-4, to extend the Funda- mental Theorem of Galois Theory from finite separable field extensions to infinite separable algebraic extensions. I hope that this new edition presents mathematics in a more natural way, making it simpler to digest and to use. I have often been asked whether solutions to exercises are available. I believe it is a good idea to have some solutions available for undergraduate students, for they are learning new ways of thinking as well as new material. Not only do solutions illustrate new techniques, but comparing them to one’s own solution also builds confidence. But I also believe that graduate students are already sufficiently confident as a result of their previous studies. As Charlie Brown in the comic strip Peanuts says, “In the book of life, the answers are not in the back.”

Acknowledgments

The following mathematicians made comments and suggestions that greatly im- proved the first two editions: Vincenzo Acciaro, Robin Chapman, Daniel R. Grayson, IlyaKapovich,T.-Y.Lam,DavidLeep,NickLoehr,RandyMcCarthy,Patrick Szuta, and Stephen Ullom. I thank them again for their help. For the present edition, I thank T.-Y. Lam, Bruce Reznick, and Stephen Ullom, who educated me about several fine points, and who supplied me with needed references. I give special thanks to Vincenzo Acciaro for his many comments, both mathe- matical and pedagogical, which are incorporated throughout the text. He carefully read the original manuscript of this text, apprising me of the gamut of my errors, from detecting mistakes, unclear passages, and gaps in proofs, to mere typos. I rewrote many pages in light of his expert advice. I am grateful for his invaluable help, and this book has benefited much from him.

Joseph Rotman Urbana, IL, 2015 . Bibliography

1. Albert, A. A., editor, Studies in Modern Algebra, MAA Studies in Mathematics, Vol. 2, Mathematical Association of America, Washington, 1963. 2. Anderson, F. W., and Fuller, K. R., Rings and Categories of Modules, 2nd ed., Springer- Verlag, Berlin, 1992. 3. Artin, E., Galois Theory, 2nd ed., Notre Dame, 1955; Dover reprint, Mineola, 1998. 4. Aschbacher, M., Lyons, R., Smith, S. D., Solomon, R., The classification of finite simple groups. Groups of characteristic 2 type, Mathematical Surveys and Monographs, 172. Amer- ican Mathematical Society, Providence, 2011.. 5. Atiyah, M., and Macdonald, I. G., Introduction to Commutative Algebra, Addison–Wesley, Reading, 1969. 6. Baker, A., Transcendental Number Theory, Cambridge University Press, Cambridge, 1975. 7. Becker, T., and Weispfenning, V., Gr¨obner Bases: A Computational Approach to Commu- tative Algebra, Springer-Verlag, New York, 1993. 8. Birkhoff, G., and Mac Lane, S., A Survey of Modern Algebra, 4th ed., Macmillan, New York, 1977. 9. Blyth, T. S., Module Theory; An Approach to Linear Algebra, Oxford University Press, 1990. 10. Borevich, Z. I., and Shafarevich, I. R., Number Theory, Academic Press, Orlando, 1966. 11.Bott,R.,andTu,L.W.,Differential Forms in Algebraic Topology, Springer-Verlag, New York, 1982. 12. Bourbaki, N., Elements of Mathematics; Algebra I; Chapters 1–3, Springer-Verlag, New York, 1989. 13. , Elements of Mathematics; Commutative Algebra, Addison–Wesley, Reading, 1972. 14. Buchberger, B., and Winkler, F., editors, Gr¨obner Bases and Applications, LMS Lecture Note Series 251, Cambridge University Press, 1998. 15. Burnside, W., The Theory of Groups of Finite Order, 2nd ed., Cambridge University Press, 1911; Dover reprint, Mineola, 1955. 16. Cajori, F., A History of Mathematical Notation, Open Court, 1928; Dover reprint, Mineola, 1993. 17. Cartan, H., and Eilenberg, S., Homological Algebra, Princeton University Press, Princeton, 1956. 18. Carter, R., Simple Groups of Lie Type, Cambridge University Press, Cambridge, 1972.

681 682 Bibliography

19. Cassels, J. W. S., and Fr¨ohlich, A., ,ThompsonBookCo.,Wash- ington, D.C., 1967. 20. Chase, S. U., Harrison, D. K., and Rosenberg, A., Galois Theory and Cohomology of Com- mutative Rings, Mem. Amer. Math. Soc. No. 52, Providence, 1965, pp. 15–33. 21. Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., and Wilson, R. A., ATLAS of Finite Groups, Oxford University Press, 1985. 22. Cox, D., Little, J., and O’Shea, D., Ideals, Varieties, and Algorithms, 2nd ed., Springer- Verlag, New York, 1997. 23. Cuoco, A.A., and Rotman, J. J., Learning Modern Algebra from Early Attempts to Prove Fermat’s Last Theorem, MAA Textbooks, Mathematical Association of America, Washing- ton D. C., 2013. 24. Dauns, J., Modules and Rings, Cambridge University Press, 1994. 25. De Meyer, F.,and Ingraham, E., Separable Algebras over Commutative Rings, Lecture Notes in Mathematics, Vol. 181, Springer-Verlag, New York, 1971. 26. Dieudonn´e, J., La Ge´ometrie des Groupes Classiques, Springer-Verlag, Berlin, 1971. 27. Dixon, J. D., du Sautoy, M. P. F., Mann, A., and Segal, D., Analytic Pro-p Groups,2nded., Cambridge University Press, 1999. 28. Dummit, D. S., and Foote, R. M., Abstract Algebra, 2nd ed., Prentice Hall, Upper Saddle River, 1999. 29. Dye, R. L. , On the Arf Invariant, Journal of Algebra 53 (1978), pp. 36–39. 30. Eisenbud, D.,Commutative Algebra with a View Toward Algebraic Geometry, Springer- Verlag, New York, 1994. 31. Finney Jr., R. L., and Rotman, J. J., Paracompactness of locally compact Hausdorff spaces, Michigan Math. J. 17 (1970), 359–361. 32. Fitchas, N., and Galligo, A., Nullstellensatz effectif et conjecture de Serre (th´eor`eme de Quillen–Suslin) pour le calcul formel, Math. Nachr. 149 (1990), 231–253. 33. Formanek, E., Central polynomials for rings, J. Algebra 23 (1972), 129–132. 34. Fr¨ohlich,A.,andTaylor,M.J.,Algebraic Number Theory, Cambridge Studies in Advanced Mathematics 27, Cambridge University Press, 1991. 35. Fuchs, L., Abelian Groups, Publishing House of the Hungarian Academy of Science, Bu- dapest, 1958. 36. Fuchs, L., Infinite Abelian Groups I, Academic Press, Orlando, 1970. 37. , Infinite Abelian Groups II, Academic Press, Orlando, 1973. 38. Fulton, W., Algebraic Curves, Benjamin, New York, 1969. 39. , Algebraic Topology; A First Course, Springer-Verlag, New York, 1995. 40. Gaal, L., Classical Galois Theory with Examples, 4th ed., Chelsea, American Mathematical Society, Providence, 1998. 41. Gorenstein, D., Lyons, R. and Solomon, R., The Classification of the Finite Simple Groups, Math. Surveys and Monographs, Vol. 40, American Mathematical Society, Providence, 1994. 42. Hadlock, C., Field Theory and Its Classical Problems, Carus Mathematical Monographs, No. 19, Mathematical Association of America, Washington, 1978. 43. Hahn, A. J., Quadratic Algebras, Clifford Algebras, and Arithmetic Witt Groups,Universi- text, Springer-Verlag, New York, 1994. 44. Hardy, G. H., and Wright, E. M., An Introduction to the Theory of ,4thed.,Oxford University Press, 1960. 45. Harris, J., Algebraic Geometry; A First Course, Springer-Verlag, New York, 1992. 46. Herrlich, H., and Strecker, G. E., Category Theory. An Introduction, Allyn & Bacon, Boston, 1973. 47.Herstein,I.N.,Topics in Algebra, 2nd ed., Wiley, New York, 1975. Bibliography 683

48. , Noncommutative Rings, Carus Mathematical Monographs, No. 15, Mathematical Association of America, Washington, 1968. 49. Hurewicz, W., and Wallman, H., Dimension Theory, Princeton University Press, Princeton, 1948. 50. Isaacs, I. M., Roots of Polynomials in Algebraic Extensions of Fields, American Mathematical Monthly 87 (1980), 543–544. 51. Jacobson, N., Basic Algebra I, Freeman, San Francisco, 1974. 52. , Basic Algebra II, Freeman, San Francisco, 1980. 53. , Lectures in Abstract Algebra III, Springer-Verlag, 1975. 54. , Structure of Rings, Colloquium Publications, 37, American Mathematical Society, Providence, 1956. 55. Kaplansky, I., Commutative Rings, University of Chicago Press, 1974. 56. , Fields and Rings, 2nd ed., University of Chicago Press, 1972. 57. , Infinite Abelian Groups, 2nd ed., University of Michigan Press, Ann Arbor, 1969. 58. , Selected Papers and Writings, Springer-Verlag, New York, 1995. 59. , Linear Algebra and Geometry; a Second Course, Allyn & Bacon, Boston, 1969. 60. , Set Theory and Metric Spaces, Chelsea, American Mathematical Society, Provi- dence, 1977. 61. Kharazishvili, Nonmeasurable Sets and Functions, North Holland Mathematics Studies 195, Elsevier, Amsterdam, 2004. 62.King,R.B.,Beyond the Quartic Equation,Birkh¨auser, Boston, 1996. 63. Kostrikin, A. I., and Shafarevich, I. R. (editors), Algebra IX. Finite Groups of Lie Type; Finite-Dimensional Division Algebras, Encyclopaedia of Mathematical Sciences, 77, Springer-Verlag, New York, 1996. 64. Lam, T. Y., The Algebraic Theory of Quadratic Forms, Benjamin, Reading, 1973, 2nd revised printing, 1980. 65. , A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991. 66. , Lectures on Modules and Rings, Springer-Verlag, New York, 1999. 67. ,Lam,T.Y.,Serre’s problem on projective modules, Springer Monographs in Math- ematics. Springer-Verlag, Berlin, 2006. 68. Lang, S., Algebra, Addison–Wesley, Reading, 1965. 69. Lyndon, R. C., and Schupp, P. E., Combinatorial Group Theory, Springer-Verlag, New York, 1977. 70. Macdonald, I. G., Algebraic Geometry; Introduction to Schemes, Benjamin, New York, 1968. 71. Mac Lane, S., Categories for the Working , Springer-Verlag, New York, 1971. 72. , Homology, Springer-Verlag, New York, 3rd corrected printing, 1975. 73. Mac Lane, S., and Birkhoff, G., Algebra, MacMillan, New York, 1967. 74. Malle, G., and Matzat, B., Inverse Galois Theory, Springer-Verlag, New York, 1999. 75. May, J. P., Munshi’s proof of the Nullstellensatz, Amer. Math. Monthly 110 (2003), 133–140. 76. McCoy, N. H., and Janusz, G. J., Introduction to Modern Algebra, 5th ed., W. C. Brown Publishers, Dubuque, Iowa, 1992. 77. Miller, W., The Maximal Order of an Element of a Finite Symmetric Group, Amer. Math. Monthly 94 (1987), 315–322. 78. Milnor, J., Introduction to Algebraic K-Theory, Annals of Mathematical Studies, No. 72, Princeton University Press, 1971. 79. Montgomery, S., and Ralston, E. W., Selected Papers on Algebra,RaymondW.BrinkSe- lected Mathematical Papers, Vol. 3, Mathematical Association of America, Washington, 1977. 684 Bibliography

80. Mumford, D., The Red Book of Varieties and Schemes. Lecture Notes in Mathematics 1358, Springer-Verlag, Berlin, 1988. 81. Munkres, J. R., Topology, A First Course, Prentice Hall, Upper Saddle River, 1975. 82. , Elements of Algebraic Topology, Addison–Wesley, Reading, 1984. 83. Navarro, G., On the fundamental theorem of finite abelian groups, Amer. Math. Monthly 110 (2003), pp. 153–154. 84. Neukirch, J., Schmidt, A., and Wingberg, K., Cohomology of Number Fields, Grundlehren der mathematischen Wissenschaften, Vol. 323, Springer-Verlag, New York, 2000. 85. Niven, I., and Zuckerman, H. S., An Introduction to the Theory of Numbers, Wiley, New York, 1972. 86. Northcott, D. G., Ideal Theory, Cambridge University Press, 1953. 87. Ol’shanskii, A. Y., Geometry of Defining Relations in Groups, Kluwer Academic Publishers, Dordrecht, 1991. 88. O’Meara, O. T., Introduction to Quadratic Forms, Springer-Verlag, New York, 1971. 89. Procesi, C., Rings with Identities, Marcel Dekker, New York, 1973. 90. Razmyslov, Ju. P., A certain problem of Kaplansky (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 483–501. 91. Rado, R., A proof of the basis theorem for finitely generated Abelian groups, J. London Math. Soc. 26 (1951), pp. 75–76, erratum, 160. 92.Robinson,D.J.S.,A Course in the Theory of Groups, 2nd ed., Springer-Verlag, New York, 1996. 93. Rosset, S., A New Proof of the Amitsur-Levitski Identity, Israel Journal of Mathematics 23, 1976, pp. 187–188. 94. Rotman, J. J., A First Course in Abstract Algebra, 3rd ed., Prentice Hall, Upper Saddle River, NJ, 2006. 95. , Galois Theory, 2nd ed., Springer-Verlag, New York, 1998. 96. , An Introduction to Homological Algebra, 2nd ed., Springer, New York, 2009. 97. , An Introduction to the Theory of Groups, 4th ed., Springer-Verlag, New York, 1995. 98. , Covering complexes with applications to algebra, Rocky Mountain J. of Math. 3 (1973), 641–674. 99. , An Introduction to Algebraic Topology, Springer-Verlag, New York, 1988. 100. , Journey into Mathematics, Prentice Hall, Upper Saddle River, 1998, Dover reprint, Mineola, 2007. 101. Rowen, L. H., Polynomial Identities in , Academic Press, New York, 1980. 102. , Ring Theory, Vols. I, II, Academic Press, Boston, 1988. 103. S¸asiada, E. Proof that every countable and reduced torsion-free abelian group is slender, Bull. Acad. Polon. Sci. 7, (1959), 143-144. 104. Serre, J.-P., Alg`ebre Locale: Multiplicit´es, Lecture Notes in Mathematics 11, 3rd ed., Springer-Verlag, New York, 1975; English transl., Local Algebra, Springer Monographs in Mathematics, Springer–Verlag, 2000. 105. , Corps Locaux, Hermann, Paris, 1968; English transl., Local Fields, Graduate Texts in Mathematics, 67, Springer–Verlag, 1979. 106. , Faisceaux alg`ebriques coh´erents, Annals Math. 61 (1955), 197-278. 107. , Topics in Galois Theory, Jones and Bartlett, Boston, 1992. 108. Shafarevich, I. R., Algebra I. Basic Notions of Algebra, Encyclopaedia of Mathematical Sciences, 11, Springer-Verlag, Berlin, 1990. 109. , Basic Algebraic Geometry, Springer-Verlag, New York, 1974. 110. Simmons, G. J., The Number of Irreducible Polynomials of Degree n over GF(p), Amer. Math. Monthly 77 (1970), pp. 743-745. Bibliography 685

111. Small, C., Arithmetic of Finite Fields, Monographs and Textbooks in Pure and Applied Mathematics 148, Marcel Dekker, Inc., New York, 1991. 112. Stewart, I., Galois Theory, 3rd ed., Chapman & Hall/CRC, Boca Raton, 2004. 113. Stillwell, J., Mathematics and Its History, Springer-Verlag, New York, 1989. 114. Suzuki, M., Group Theory I, Springer-Verlag, New York, 1982. 115. Tignol, J.-P., Galois’ Theory of Algebraic Equations, World Scientific Publishing Co., Inc., River Edge, 2001. 116. van der Waerden, B. L., Geometry and Algebra in Ancient Civilizations, Springer-Verlag, New York, 1983. 117. , A , Springer-Verlag, New York, 1985. 118. , Modern Algebra, Vols. I, II, 4th ed., Ungar, New York, 1966. 119. , Science Awakening, Wiley, New York, 1963. 120. Weyl, H., The Classical Groups; Their Invariants and Representations, Princeton, 1946. 121. Williams, K. S., Note on Non-Euclidean Domains, Math. Mag., 48 (1975), pp. 176–177. 122. Zariski, O., and Samuel, P., Commutative Algebra I, van Nostrand, Princeton, 1958. 123. , Commutative Algebra II, van Nostrand, Princeton, 1960.

Special Notation

|X| cardinal number of set X C complex numbers N natural numbers Q rational numbers R real numbers Z integers 1 identity function on set XA transpose of matrix A X n n! Z r binomial coefficient r!(n−r)! m integers mod m

Course I √ − 1 3 ω : cube : ω = 2 + i 2 ...... 5 a | b : a is a of b ...... 9 gcd(a, b) : greatestcommondivisor ...... 10 a ≡ b mod m : a and b congruent mod m ...... 19

Matn(R): n × n matrices,realentries ...... 30

δij : Kronecker delta: δij =0ifi = j and δii =1 ...... 30 End(V ) : endomorphismring ...... 31

Zm : integers mod m ...... 31 Z[i] : Gaussianintegers...... 32 Z[ω] : Eisensteinintegers ...... 32 2X : Boolean ring of all subsets of X ...... 33 F(X) : real-valued functions on X ...... 34 C(X): allcontinuous real-valued functions on a space X ...... 35 C∞(X): allf : X → R having all nthderivatives...... 35 U(R) : group of units of R ...... 37

Fp : finite field with p elements; another name for Zp ...... 37 Frac(R) : field of R ...... 38

687 688 Special Notation

Ac : set-theoretic complement of subset A ...... 40 deg(f) : degree of a polynomial f(x) ...... 42 R[x] : polynomial ring over a commutative ring R ...... 42 R[[x]] : formal power series ring over a commutative ring R ...... 42 x : indeterminate...... 43 f  : polynomial function R → R of f(x) ∈ R[x] ...... 44 k(x) : field of rational functions of field k ...... 44 ∼ A = R : isomorphism of rings A and R ...... 47 ea : R[x] → R : evaluation at a ∈ R ...... 49

(b1,...,bn) : ideal generated by b1,...,bn ...... 51 (b) : principal ideal generated by b ...... 51 R × S : direct product of rings R and S ...... 54 a + I : coset of ideal I with representative a ∈ R ...... 55 R/I : quotient ring of ring R mod ideal I ...... 55 ϕ−1(S) : inverse image of S ⊆ Y if ϕ: X → Y ...... 61 lcm(f,g) : of f and g ...... 72 K/k : K is an extension field of a field k ...... 78 [K : k] : degree of extension field K/k ...... 78 k(α) : field obtained from k by adjoining α ...... 79 irr(α, k) : minimal polynomial of α over field k ...... 80 μ(m): M¨obiusfunction ...... 86 n Fq : finite field with q = p elements...... 88

Φd(x) : cyclotomicpolynomial...... 93 ∂ : degreefunctionofeuclideanring ...... 98 PID: principalidealdomain ...... 101 UFD: uniquefactorizationdomain...... 104

SX : symmetric group on set X ...... 116

Sn : symmetric group on n letters ...... 117

(i1,i2 ...,ir): r-cycle ...... 117 sgn(σ) : signum of permutation σ ...... 125 GL(n, k) : general linear group over commutative ring k ...... 128 B(X) : Boolean group on set X ...... 129 S1 : circlegroup ...... 129

Γn :groupofnthrootsofunity ...... 129 |G| : order of group G ...... 135

D2n : dihedral group of order 2n ...... 137 V : four-group ...... 137 Special Notation 689

A : alternatinggroup ...... 141 n a : cyclic subgroup generated by a ...... 141 φ(n) : Euler φ-function...... 142 H ∨ K : subgroup generated by subgroups H and K ...... 143 aH : (multiplicative) left coset of subgroup H ...... 144 [G : H] : index of subgroup H ⊆ G ...... 147 K  G : K is a normal subgroup of G ...... 153

γg : G → G : conjugation by g ∈ G ...... 154 Z(G) : center of group G ...... 155 Aut(G) : automorphism group of group G ...... 155 Inn(G) : inner automorphism group of group G ...... 155 Q : quaterniongroupoforder8 ...... 156 G/K : quotient group G by normal subgroup K ...... 161 H × K : direct product of groups H and K ...... 167 G : commutator subgroup of group G ...... 172 Gal(E/k) : Galois group of extension field E/k ...... 181 Fr : k → k : Frobenius automorphism of field k of characteristic p ...... 186 EH :fixedfieldofH ⊆ Gal(E/k) ...... 202 A ∨ B : compositum of subfields A and B ...... 209 x  y : x precedes y ina partialorder ...... 209 dimk(V ), dim(V ) : dimension of vector space V over k ...... 255

Homk(V,W): allk-linear transformations V → W ...... 260 V ∗ : dual space of vector space V ...... 260 GL(V ) : general linear group of vector space V ...... 260

Y [T ]X : matrix of transformation T with respect to bases Y and X ...... 264

I, In :identitymatrix,n × n identitymatrix ...... 264 Course II kG : group algebra: G group, k commutativering ...... 274 ZG : group ring of group G ...... 274

Endk(M) : endomorphism ring of k-module ...... 275 Z(R) : center of ring R ...... 277

RM : M is a left R-module ...... 289

MR : M is a right R-module ...... 289

μr : multiplication by r ...... 291 Rop : oppositering ...... 292 x : cyclic module with generator x ...... 296 coker ϕ : cokernel of map ϕ ...... 297 690 Special Notation

supp(v): supportofelementv withrespecttobasis ...... 322 A : directproduct ...... 323 .i∈I i i∈I Ai : directsum...... 323 G : augmentation ideal of group G ...... 338 A : allalgebraicnumbers...... 340 k : algebraic closure of field k ...... 341 trdeg(E/k) : transcendencedegree ...... 351 height(ϕ) : height of rational function ϕ ...... 353 tG : torsion subgroup of G ...... 359

Gp : p-primarycomponent ...... 362 ann(m) : order ideal of element m ...... 379 ann(M) : annihilator of module M ...... 381 C(g) : companion matrix of polynomial g ...... 385 det(A) : of matrix A ...... 389

ψA(x) : characteristic polynomial of matrix A ...... 390 tr(A) : trace of matrix A ...... 390 J(α, s) : Jordanblock...... 395 PSL(n, k) : projectiveunimodulargroup ...... 402 (X | Y ) : presentation ...... 403 (V,f) : innerproductspace ...... 417 W ⊥ : orthogonal complement ...... 421

W1⊥W2 : orthogonal direct sum ...... 424 Sp(V,f) : symplecticgroup...... 431 O(V,f) : orthogonal group ...... 431 obj(C) : class of objects in category C ...... 443 Hom(A, B) : setofmorphisms...... 443

1A : identitymorphism...... 443 Sets : categoryofsets ...... 444 Groups : categoryofgroups ...... 444 Ab : categoryofabeliangroups ...... 444 Rings : categoryofrings...... 444 ComRings : categoryofcommutativerings...... 444

RMod : category of left R-modules ...... 444

ModR : category of right R-modules ...... 444 PO(X) : categoryofpartiallyorderedset ...... 444 C(G) : category of group G ...... 445 hTop : homotopycategory...... 445 Special Notation 691

A B : coproduct...... 447 A B : categoricalproduct ...... 450 Hom(A, ) : covarianthomfunctor ...... 462

Top ∗ : categoryofpointedspaces ...... 463 Cop : oppositecategory...... 465

RMS :(R, S)-bimodule...... 470 IBN: invariantbasisnumber ...... 483 dG : maximal divisible subgroup of G ...... 502 Z∞ p :Pr¨ufergroup ...... 503 δ∞(D): dimQ(D), where D isdivisibleabeliangroup...... 505

A ⊗k B : tensor product of k-modules ...... 510   f ⊗ g :mapA ⊗k B → A ⊗k B ...... 512

A ⊗k B : tensor product of k-algebras ...... 546 Ae : envelopingalgebra ...... 548 3 p M : tensor product of module M with itself p times ...... 555 T (M) : tensor algebra on module M ...... 555 Gr Alg : category of graded k-algebras ...... 556 4 k M : exterior algebra on k-module M ...... 562 4 m ∧ m : wedge product in M of m, m ∈ M ...... 562 4 p M : pth exterior power of k-module M ...... 562 ∧···∧ ≤ ≤···≤ ≤ eH : ei1 eip for H =1 i1 ip n ...... 566 G(V ) : Grassmann algebra on free k-module V ...... 566 det(f) : determinant of f : M → M, ...... 576 det(A) : determinant of matrix A ...... 576

AH,L : submatrix ...... 581

Aij :matrixwithrowi and column j deleted ...... 584 adj(A) : adjointmatrix...... 584

Cij : cofactor ...... 584 f(X): f(x1,...,xn) ...... 593 f  : associatedpolynomialfunction...... 593 Var(I) : affine variety of ideal I ...... 594 Id(V ) : idealofvariety ...... 597 k[A] : coordinate ring of A ⊆ kn ...... 597 √ I : radical of ideal I ...... 598 (I : S) : colonideal...... 603 k[a1,...,an] : finitely generated k-algebra ...... 604

OR/k : integral elements in k-algebra R ...... 604 692 Special Notation

nil(k) : nilradical of commutative ring k ...... 608 Spec(k) : set of prime ideals in commutative ring k ...... 615 aR/Q : R/Q → R/Q :mapr + Q → ar + Q if Q is ideal in R ...... 618 k(V ) : coordinate field k(V )=Frac(k[V ]) if V isirreducible ...... 625 Aff(k) : category of affine varieties over k ...... 627 cXα : monomialinseveralvariables ...... 630 deg : degreeofmonomial ...... 630 |α| : weightofmonomial ...... 630 LM(f) : leadingmonomial ...... 631 W +(Ω): monoidofwordsonwell-orderedsetΩ ...... 632 S(f,g): S-polynomial...... 641 { j } Mi,ψi : inversesystem ...... 651 lim M : inverselimit ...... 653 ←−j i Z∗ p : p-adicintegers...... 655 { i } Mi,ϕj : directsystem...... 657 lim M : directlimit ...... 658 −→j i Index

Abel, N. H., 7, 219 numbers, 340 abelian group, 128 algebraically divisible, 496 closed, 341 free, 328 dependent, 345 primary, 362 algorithm reduced, 502 Buchberger, 646 torsion, 380 Euclidean, 17 torsion-free, 380 into disjoint cycles, 118 absolute Galois group, 480 , 319 ACC, 282, 300 alternating Accessory Irrationalities, 199 bilinear form, 418 action of group, 152 group, 141 transitive, 187 multilinear, 563 additive functor, 465 space, 418 additive notation, 130 alternating sum, 26 Adelard of Bath, 4 Amitsur, S. A., 560 adjacency, 127 annihilator adjoining a unit, 39 element, 379 adjoining to field, 79 module, 381 adjoint antanairesis, 16 functors, 666 anti-isomorphism, 293 linear transformation, 431, 437 Apollonius, 4 matrix, 584 , 4 Adjoint Isomorphism, 526, 527 Arf invariant, 429 affine group, 139 Arf, C., 429 affine variety, 594 artinian ring, 286 algebra, 284, 543 ascending chain condition, 282 enveloping, 548 associated polynomial function, 593 finitely generated, 604 associated prime ideal, 620 generated by n elements, 558 associates, 52 graded, 550 associativity, 29, 128 algebra map, 543 generalized, 131, 553 algebraic augmentation, 338 closure, 341 augmentation ideal, 338 element, 79 automorphism extension, 79 field, 180

693 694 Index

group, 155 Carnap, R., 461 inner, outer, 155 Cartan, H., 538 automorphism group, 155 Cartan–Eilenberg Theorem, 538 Axiom of Choice, 313 cartesian product, 235 castle problem, 8 b-adic digits, 23 , 189 Baer Theorem, 537 category, 443 Baer, R., 494 composition, 443 base b,23 morphism, 443 base of topology, 675 objects, 443 basepoint, 463 opposite, 465 basic subgroup, 521 pre-additive, 446 basis small, 525 dependence, 349 Cauchy sequence, 654 free abelian group, 328 Cauchy, A.-L., 7 free algebra, 556 Cayley, A., 140 free module, 329, 481 Cayley–Hamilton Theorem, 392 ideal, 283 center standard, 253 group, 155 vector space matrix ring, 268, 281 finite-dimensional, 252 ring, 277 infinite-dimensional, 319 centerless, 155 Basis Theorem chain, 314 finite abelian groups, 367, 499 change of rings, 475 Hilbert, 286 character, 203 Bass, H., 300, 498 character group, 532 Beltrami, E., 594 characteristic of field, 60 biadditive, 509 characteristic polynomial, 390 bifunctor, 521 Ch’in Chiu-shao, 8 bijection, 241 Chinese Theorem bilinear form, 417 Z,25 alternating, 418 k[x], 89 nondegenerate, 420 circle operation, 280 skew, 418 circle group, 129 symmetric, 417 class group, 540 negative definite, 426 Classification Theorem of Finite Simple positive definite, 426 Groups, 176 bilinear function, 417, 509 Clifford algebra, 572 bimodule, 470 Clifford, W. K., 572 , 29 coefficients, 41 Binomial Theorem cofactor, 584 commutative ring, 32 cofinal subset, 318 exterior algebra, 569 cofinite, 41, 596 birational map, 627 Cohen, I. S., 317 Bkouche, R., 488 cokernel, 297 Boole, G., 129 colimit (see direct limit), 658 Boolean group, 129 colon ideal, 603 Boolean ring, 33, 41 Columbus, 4 Buchberger’s algorithm, 646 column space of matrix, 270 Buchberger’s Theorem, 643 commensurable, 13 Buchberger, B., 629, 640 common divisor, 10 in Z,10 C∞-function, 35 several polynomials, 103 cancellation law two polynomials, 66 domain, 34 commutative, 128 group, 130 commutative diagram, 305 Cardano, G., 5 commutative ring, 32 Index 695

domain, 34 ideal, 55 euclidean ring, 98 subgroup, 144 factorial, 104 covariant functor, 461 field, 37 Cramer’s Rule, 586 Jacobson, 610 Cubic Formula, 5 PID, 101 cubic polynomial, 44, 188 polynomial ring, 42 cycle several variables, 45 permutation, 117 reduced, 598 cycle structure, 120 UFD, 105 cyclic compact, 674 group, 141 companion matrix, 385 module, 296 complement, 40, 325 cyclotomic polynomial, 93 complete factorization, 120 completion, 655 DCC, 286, 301 complex De Morgan laws, 41 de Rham, 574 De Morgan, A., 41 modulus, 129 de Rham complex, 574 composite , 11 de Rham, G., 574 composite of functions, 239 Dean,R.A.,39 composition factors, 195 Dedekind ring, 535 composition series, 195, 302 Dedekind Theorem, 204 length, 195 Dedekind, R., 204 composition, category, 443 degree compositum, 209 euclidean ring, 98 congruence mod I,55 extension field, 78 congruence class, 244 graded map, 550 congruent mod m,19 homogeneous element, 550 congruent matrices, 419 polynomial, 42 conjugacy class, 157 several variables, 631 conjugate several variables, 630 group elements, 154 degree-lexicographic order, 634 intermediate fields, 207 derivation, 587 conjugation , 46 Grassmann algebra, 567 Descartes, R., 3, 7 groups, 154 determinant, 576 , 276 diagonalizable, 394, 401 constant function, 236 diagram, 305 constant functor, 462 commutative, 305 constant polynomial, 44 diagram chasing, 308 constant term, 44 Dickson, L. E., 122 content, 109 Dieudonn´e, J., 558 continuous, 675 differential form, 574 contravariant functor, 464 dihedral group, 136 convolution, 274, 282 dimension, 255, 322 coordinate field, 625 , 4 coordinate list, 253 direct limit, 658 coordinate ring, 597 direct product Copernicus, 4 commutative rings, 54 coproduct groups, 167 family of objects, 452 modules, 323, 451 two objects, 447 rings, 275 Correspondence Theorem direct sum groups, 165 matrices, 384 modules, 298 modules, 323, 324, 451 rings, 279 external, 324, 326 coset internal, 326 696 Index

vector spaces, 259, 268 equivalent direct summand, 325 filtration, 302 direct system, 657 matrices, 406 transformation, 662 normal series, 197 directed set, 659 series, groups, 197 Dirichlet, J. P. G. L., 368 , 4 discrete, 678 etymology discriminant, 223 abelian, 219 bilinear form, 420 adjoint functors, 666 of cubic, 224 affine, 594 of quartic, 230 affine space, 627 disjoint permutations, 117 alternating group, 141 disjoint union, 452 automorphism, 155 distributivity, 29 canonical form, 386 divides commutative diagram, 305 commutative ring, 36 cubic, 44 in Z,9 cycle, 117 divisible module, 496 dihedral group, 136 Division Algorithm domain, 34 k[x], 62 exact sequence, 575 k[x1,...,xn], 637 exterior algebra, 562 in Z,10 field, 37 division ring, 275 flat, 529 divisor functor, 461 in Z,9 homomorphism, 47 domain isomorphism, 47 commutative ring, 34 kernel, 50 morphism, 443 left exact, 469 of function, 236 polyhedron, 136 PID, 101 power, 130 UFD, 105 profinite, 477 dual basis, 269 pure subgroup, 364 dual space, 260, 269 quadratic, 44 duals in category, 450 quasicyclic, 503 Dye, R. L., 429 quaternions, 276 quotient group, 162 eigenvalue, 388 radical, 598 eigenvector, 388 rational canonical form, 386 Eilenberg, S., 441, 491, 538 ring, 29 Eisenstein Criterion, 95 symplectic, 424 Eisenstein integers, 32 torsion subgroup, 359 Eisenstein, G., 95 variety, 594 elementary divisors vector, 248 finite abelian group, 373 , 4 matrix, 397 Euclid’s Lemma, 69, 98, 101 elementary matrix, 410 integers, 12 elimination ideal, 648 Euclidean Algorithm I endomorphism integers, 17 abelian group, 274 Euclidean Algorithm II module, 294 integers, 18 ring, 274 Euclidean Algorithm, k[x], 70 enlargement of coset, 62, 165, 298 euclidean ring, 98 enveloping algebra, 548 Eudoxus, 4 equal subsets, 236 Euler φ-function, 142 equality of functions, 118 Euler Theorem, 148 equivalence class, 244 Euler, L., 19 equivalence relation, 243 evaluation homomorphism, 49 Index 697

even permutation, 124 finite exact extension, 78 functor, 469 order (module), 379 left, 467 topology, 479, 679 right, 517 finite index topology, 675 sequence, 305 finite-dimensional, 251 factored, 310 finitely generated short, 306 algebra, 604 splice, 310 ideal, 283 Exchange Lemma, 256 module, 296 exponent finitely presented module, 488 group, 376 Finney, Jr., R. L., 488 module, 381 First Isomorphism Theorem extension commutative rings, 58 modules, 306 groups, 163 extension field, 78 modules, 297 algebraic, 79 vector spaces, 269 degree, 78 Five Lemma, 309 finite, 78 fixed field, 202 Galois, 207, 475 fixes, 117, 180 inseparable, 182 flat module, 529 normal, 190 forgetful functor, 462 pure, 187 formal power series purely transcendental, 345 one variable, 41 radical, 187 Formanek, E., 560 separable, 182 four-group, 137 simple, 214 fraction field, 38 exterior algebra, 562 fractional ideal, 539 exterior derivative, 574 Fraenkel, A. A. H., 442 exterior power, 562 free abelian group, 328 factor groups, 192 algebra, 556 factor modules, 302 commutative algebra, 558, 671 factorial ring (see UFD), 104 module, 329, 481 faithful module, 292 freeness property, 330 Feit, W., 219 Frobenius Feit–Thompson Theorem, 219 automorphism, 186 Fermat Little Theorem, 22 Frobenius, G., 374 Fermat prime, 96 function, 236 Fermat’s Theorem, 148 bijection, 241 Ferrari, Lodovici, 5 constant, 236 Fibonacci, 4, 590 identity, 236 field, 37 inclusion, 237 algebraic closure, 341 injective, 238 algebraically closed, 341 polynomial, 44 finite, 186 rational, 45 fraction, 38 restriction, 239 Galois, 88 surjective, 238 perfect, 401 functor prime, 59 additive, 465 rational functions, 44 constant, 462 15-puzzle, 124, 126 contravariant, 464 filtration, 302 contravariant Hom, 464 length, 302 covariant, 461 refinement, 302 covariant Hom, 461 filtrations exact, 469 equivalent, 302 forgetful, 462 698 Index

identity, 461 Grassmann algebra, 566 left exact, 467, 468 Grassmann, H. G., 566 representable, 528 right exact, 517 domain, 97 two variables, 521 in Z,10 fundamental group, 463 several polynomials, 103 Fundamental Theorem two polynomials, 66 Arithmetic, 198 Gr¨obner, W., 640 finite abelian groups Gr¨obner basis, 640 elementary divisors, 374 Grothendieck, A., 441, 592 invariant factors, 376 group finitely generated abelian groups abelian, 128 elementary divisors, 374 additive notation, 130 invariant factors, 377 affine, 139 Galois Theory, 211, 479 algebra, 274 modules alternating, 141 elementary divisors, 382 axioms, 128, 138 invariant factors, 382 Boolean, 129 symmetric functions, 208 circle group, 129 symmetric polynomials, 208, 639 conjugacy class, 157 cyclic, 141 G-domain, 606 dihedral, 136 G-ideal, 608 four-group, 137 Galligo, A., 487 free abelian, 328 Galois extension, 207, 475 Galois, 181 Galois field, 88 general linear, 128 Galois group, 181, 475 hamiltonian, 156 absolute, 480 modular, 173 Galois Theorem, 86 Pr¨ufer, 503 Galois, E., 8, 146 quasicyclic, 503 Gauss Theorem quaternions, 156 R[x] UFD, 110 quotient, 162 cyclotomic polynomial, 96 simple, 173 Gauss’s Lemma, 111 solvable, 192 Gauss, C. F., 215 special linear, 140 Gaussian elimination, 409 special unitary, 437 Gaussian equivalent, 410 stochastic, 139 Gaussian integers, 32 symmetric, 117, 128 gcd, 10 topological, 461, 678 Gelfond, A., 347 torsion, 359 Gelfond-Schneider Theorem, 347 torsion-free, 359 general linear group, 128 unitary, 437 general polynomial, 84 group algebra, 274 Generalized Associativity, 131 group object, 460 generalized associativity, 553 group of units, 37 generate Gruenberg, K. A., 481 dependence, 349 Gutenberg, 4 generator cyclic group, 141 Hamel basis, 321 generators and relations, 403 Hamel, G. K. W., 321 algebra, 556 Hamilton, W. R., 156, 276, 392 Gerard of Cremona, 4 hamiltonian, 156 Goldman, O., 604 Hasse, H., 429 Goodwillie, T. G., 590 Hasse–Minkowski Theorem, 429 Gordan, P., 285 Hausdorff, 676 graded algebra, 550 Hausdorff, F., 676 graded map, 550 height (rational function), 353 Index 699

Hermite, C., 122 group element, 128 hermitian, 437 morphism, 443 Hilbert, D., 29, 232, 285 image Basis Theorem, 286 function, 236 Nullstellensatz, 600, 612 linear transformation, 260 Theorem 90, 217 module homomorphism, 296 , 4 inclusion, 237 H¨older, O., 198 increasing p ≤ n list, 565 Hom functor indecomposable, 333 contravariant, 464 Independence of Characters, 203 covariant, 461 independent list, 252 homogeneous element, 550 maximal, 257 homogeneous ideal, 550 indeterminate, 43 homomorphism index of subgroup, 147 R-homomorphism, 291 induced map, 461, 464 algebra, 543 induced topology, 676 commutative ring, 47 induction (transfinite), 345 graded algebra, 550 infinite order, 133, 379 group, 150 infinite-dimensional, 251 conjugation, 154 initial object, 459 natural map, 162 injections ring, 279 coproduct, 447, 452 Houston, E., 218 direct sum of modules, 327 Hume, J., 3 injective, 238 Hurewicz, W., 305 limit (see direct limit), 658 hyperbolic plane, 424 module, 492 hypersurface, 596 inner automorphism, 155 inner product, 417 IBN, 483 matrix, 419 ideal, 50, 278 space, 417 augmentation, 338 inseparable basis of, 283 extension, 182 colon, 603 polynomial, 182 commutative ring, 50 integers, 9 elimination, 648 integers mod m,31 finitely generated, 283 integral closure, 604 fractional, 539 (see domain), 34 generated by subset, 53 intermediate field, 207 homogeneous, 550 Invariance of Dimension, 255, 256 invertible, 539 invariant (of group), 152 left, 278 invariant basis number, 483 maximal, 74 invariant factors minimal left, 287 finite abelian group, 376 monomial, 645 matrix, 386 nilpotent, 614 invariant subspace, 295 order, 379 inverse primary, 617 commutative ring, 36 prime, 75 function, 241 principal, 51 Galois problem, 232 proper, 50 group element, 128 radical, 598 image, 61 right, 278 limit, 653 two-sided, 278 right, 282 ideal generated by X, 280 system, 651 identity invertible ideal, 539 function, 236 , 585 functor, 461 irreducible 700 Index

element, 67 Laurent, P. A., 281 module (see simple module), 299 law of inertia, 427 variety, 614 Law of Substitution, 128, 237 irredundant, 620 laws of exponents, 132 union, 616 Lazard, M., 666 Isaacs, I. M., 343 leading coefficient, 42 isometry, 135, 429 least common multiple isomorphic commutative ring, 72 commutative rings, 47 in Z,14 groups, 150 Least Integer Axiom, 9 modules, 291 left exact functor, 467 isomorphism left hereditary ring, 535 R-isomorphism, 291 left noetherian ring, 284 category, 445 length groups, 150 composition series, 195 modules, 291 cycle, 117 rings, 47 filtration, 302 vector spaces, 259 module, 303 normal series, 192 Jacobson ring, 610 Leonardo da Pisa (Fibonacci), 4 Jacobson, N., 610 Levitzki, J., 560 Janusz,G.J.,222 lexicographic order, 631 Jordan canonical form, 397 lifting, 483 Jordan, C., 198 limit (see inverse limit), 653 Jordan–H¨older Theorem Lindemann, F., 347 groups, 198 linear modules, 303 fractional transformation, 353 functional, 473 k-algebra, 543 polynomial, 44 k-linear combination, 250 transformation, 259 k-map, 343 nonsingular, 259 Kaplansky Theorem, 535 linear combination Kaplansky, I., 52, 282, 560 in Z,10 kernel module, 296 group homomorphism, 153 vector space, 250 linear transformation, 260 linearly dependent list, 252 module homomorphism, 296 linearly independent infinite set, 319 ring homomorphism, 50, 279 linearly independent list, 252 Kronecker delta, 30 list, 250 Kronecker product, 520 coordinate, 253 Kronecker Theorem, 83 increasing p ≤ n, 565 Kronecker, L., 374 linearly dependent, 252 Krull Theorem, 609 linearly independent, 252 Krull, W., 318, 479 Lodovici Ferrari, 7 Kulikov, L. Yu., 521 Lo´ s, J., 454 Kurosh, A. G., 448 lowest terms in Q,12 Lagrange Theorem, 146 in k[x], 69 Lagrange, J.-L., 7, 146 L¨uroth, J., 355 Lambek, J., 533 L¨uroth’s Theorem, 355 Landau, E., 139 Luther, M., 4 Laplace expansion, 583 Laplace, P.-S., 583 m-adic topology, 676 Lasker, E., 620 Mac Lane, S., 441, 461, 553 Latin square, 157 mapping problem, universal, 449 lattice, 210 Maschke’s Theorem, 337 Laurent polynomials, 281 Maschke, H., 337 Index 701

matrix Moore, J., 491 elementary, 410 morphism, 443 linear transformation, 263 identity, 443 nilpotent, 401 Motzkin, T. S., 101 nonsingular, 128 moves, 117 scalar, 158, 268 multilinear function, 552 strictly triangular, 269 alternating, 563 maximal element multiplication by r, 291 poset, 314 multiplication table, 150 maximal ideal, 74 multiplicity, 72 maximal independent list, 257 Munshi, R., 613 maximum condition, 283 metric space, 673 natural minimal isomorphism, 523 left ideal, 287 transformation, 523 polynomial natural map, 57 matrix, 393 groups, 162 prime ideal, 318 modules, 297 minimal polynomial rings, 279 algebraic element, 80 vector spaces, 269 minimum condition, 287 natural numbers, 9, 141 Minkowski, H., 429 Navarro, G., 369 minor, 581 Niccol`o Fontana (Tartaglia), 4 M¨obius, A. F., 86 nilpotent modular group, 173 element, 598 modular law, 300 matrix, 401 module, 288 nilpotent ideal, 614 bimodule, 470 nilradical, 608 cyclic, 296 Nobeling, G., 537 divisible, 496 Noether, E., 163, 284, 620 faithful, 292 noetherian, 284, 301 finitely generated, 296 nondegenerate, 420 finitely presented, 488 quadratic form, 429 flat, 529 nonderogatory, 394 free, 329, 481 nonsingular injective, 492 linear transformation, 259 left, 288 matrix, 128 primary, 381 nontrivial subgroup, 139 projective, 484 , 216 quotient, 297 euclidean ring, 98 right, 289 normal simple, 299 extension, 190 torsion, 380 series, 192 torsion-free, 359, 380 factor groups, 192 modulus, 129 length, 192 Molien, T., 338 refinement, 197 monic polynomial, 42 subgroup, 153 several variables, 631 generated by X, 158 monkey, 27 Nullstellensatz, 600, 612 , 133 weak, 599, 612 W +(Ω), 632 monomial ideal, 645 objects of category, 443 monomial order, 630 odd permutation, 124, 126 degree-lexicographic order, 634 Ol’shanskii, A. Yu., 508 lexicographic order, 631 one-to-one Moore Theorem, 88 (injective function), 238 Moore, E. H., 88 one-to-one correspondence 702 Index

(bijection), 241 n variables, 45 onto function commuting variables, 559 (surjective function), 238 cyclotomic, 93 opposite category, 465 function, 593 opposite ring, 292 general, 84 order irreducible, 67 group, 135 monic, 42 group element, 133 noncommuting variables, 556 power series, 46 reduced, 224 order ideal, 300, 379 separable, 182 order-reversing, 210 skew, 275 ordered pair, 235 zero, 42 orthogonal polynomial function, 44, 593 basis, 425 polynomial identity, 560 complement, 421 Pontrjagin duality, 501 direct sum, 424 Pontrjagin, L. S., 333 group, 431 poset, 209, 314 matrix, 158 positive definite, 426 orthonormal basis, 425 power series, 41 outer automorphism, 155 powers, 130 Pr¨ufer, H., 365 p-adic topology, 675 pre-additive category, 446 p-adic integers, 655 presheaf, 671 p-adic numbers, 655 primary component, 362, 381 p-primary abelian group, 362 Primary Decomposition (p)-primary module, 381 commutative rings, 620 pairwise disjoint, 245 irredundant, 620 Papp, Z., 498 primary decomposition, 362 Pappus, 4 primary ideal, 617 , 248 belongs to prime ideal, 618 parity, 19, 124 prime element, 105 partially ordered set, 209 prime factorization chain, 314 in Z,11 directed set, 659 polynomial, 72 discrete, 652 prime field, 59 well-ordered, 316 prime ideal, 75 partition, 55, 245 associated, 620 partition of n, 377 belongs to primary ideal, 618 perfect field, 401 minimal, 318 permutation, 116 primitive adjacency, 127 element, 66 complete factorization, 120 theorem, 214 cycle, 117 polynomial, 108 disjoint, 117 associated, 109 even, 124 root of unity, 92 odd, 124, 126 primitive element, 85 parity, 124 principal signum, 125 ideal, 51 transposition, 117 ideal domain, 101 φ-function, 142 product PI-algebra, 560 categorical PID, 101 family of objects, 452 Pigeonhole Principle, 261 two objects, 450 Poincar´e, H., 150 direct pointed spaces, 463 groups, 167 pointwise operations, 35 modules, 323, 451 polynomial, 42 rings, 275 Index 703

product topology, 678 R-homomorphism, 291 profinite completion, 656 R-isomorphism, 291 profinite group, 680 R-linear combination, 296 projections R-map, 291 direct sum of modules, 327 R-module, 288 product, 450, 452 Rabinowitz trick, 600 projective radical extension, 187 limit (see inverse limit), 653 radical ideal, 598 module, 484 Rado, R., 369 projective unimodular group, 402 rank proper free abelian group, 329 class, 442 free module, 482 divisor, 106 linear transformation, 269 ideal, 50 matrix, 270 subgroup, 139 rational canonical form, 386 submodule, 295 rational curve, 625 subring, 32 rational functions, 44 subset, 237 rational map, 626 subspace, 249 Razmyslov, Yu. P., 560 Pr¨ufer, H., 503 Recorde, R., 3 Pr¨ufer group, 503 reduced Pr¨ufer topology, 676 abelian group, 502 pullback, 455 basis, 648 pure commutative ring, 598 extension, 187 mod {g1,...,gm}, 636 subgroup, 364 polynomial, 224 submodule, 370 reduction, 636 purely transcendental, 345 refinement, 197, 302 pushout, 456 reflexive relation, 243 , 15, 623 regular map, 626 primitive, 15 Reisz Representation Theorem, 422 Pythagorus, 4 Reisz, M., 422 relation, 243 , 8 relatively prime quadratic form, 428 k[x], 69 equivalence, 429 in Z,12 nondegenerate, 429 integers, 12 quadratic polynomial, 44 UFD, 107 Quartic Formula, 7 remainder, 10 quartic polynomial, 44, 189 k[x], 63 resolvent cubic, 7 k[x1,...,xn], 637 quasicyclic group, 503 mod G, 637 quasiordered set, 445 repeated roots, 74 quaternions, 156 representable functor, 528 division ring, 276 representation of ring, 292 Quillen, D., 487 representative of coset, 144 quintic polynomial, 44 resolvent cubic, 7, 229 quotient restriction, 239 (Division Algorithm) resultant, 225 k[x], 63 retract, 325 (Division Algorithm) in Z,10 retraction, 325 group, 162 right R-module, 289 module, 297 right exact functor, 518 space, 258 ring, 29, 273 quotient ring, 57, 278 artinian, 286 Boolean, 33, 41 r-cycle, 117 commutative, 32 704 Index

Dedekind, 535 composition, 302 division ring, 275 factor modules, 302 quaternions, 276 Serre, J.-P., 441, 487, 592 endomorphism ring, 274 sesquilinear, 436 group algebra, 274 set, 442 Jacobson, 610 sgn, 125 left hereditary, 535 Shafarevich, I., 232 left noetherian, 284 short exact sequence, 306 opposite, 292 split, 307 polynomial, 42 shuffle, 571 self-injective, 499 signature, 427 semisimple, 335 signum, 125 skew polynomial, 42 similar matrices, 154, 267 unique factorization domain, 541 Simmons, G. J., 86 zero, 31 simple root extension, 214 multiplicity, 72 group, 173 polynomial, 64 module, 299, 334 root of unity, 92, 129 transcendental extension, 353 primitive, 92 Singer, R., 95 Rosset, S., 560 single-valued, 237 Rotman, J. J., 488 skew field, 275 Ruffini, P., 7 skew polynomial ring, 42 Russell paradox, 442 skew polynomials, 275 Russell, B. A. W., 442 slender, 454 small category, 525 Sarges, H., 286 small class (= set), 442 S¸asiada, E., 454 Small, L., 288, 535 scalar smallest matrix, 158, 268 element in partially ordered set, 316 multiplication, 247 subspace, 250 module, 288 Smith normal form, 411 transformation, 268 Smith, H. J. S., 411 Schanuel’s Lemma, 489 solution dual, 500 linear system, 249 Schanuel, S., 351 universal mapping problem, 449 Schering, E., 374 solution space, 144, 249 Schneider, T., 347 solvable Schottenfels, I. M., 402 by radicals, 188 Schreier Refinement Theorem group, 192 groups, 197 spans, 250 modules, 302 infinite-dimensional space, 319 Scipio del Ferro, 4 Spec(R) Second Isomorphism Theorem topological space, 615 groups, 164 special linear group, 140 modules, 297 special unitary group, 437 secondary matrices, 417 Specker, E., 537 self-adjoint, 436 splice, 310 self-injective, 499 split short exact sequence, 307 semigroup, 133 splits semisimple module, 334 polynomial, 72, 84 semisimple ring, 335 splitting field separable polynomial, 84 element, 182 S-polynomial, 641 extension, 182 squarefree integer, 15 polynomial, 182 stalk, 671 series standard basis, 253 Index 705

standard polynomial, 560 basis, 424 Stasheff, J., 553 group, 431 Steinitz Theorem, 214 Steinitz, E., 214 tangent half- formula, 624 Stevin, S., 3 target, 236, 443, 463 Stickelberger, L., 374 Tarski monsters, 508 string, 373 Tarski, A., 508 subbase of topology, 675 Tartaglia, 4 subcategory, 446 tensor algebra, 556 subfield, 38 tensor product, 510 generated by X,59 terminal object, 459 prime field, 59 , 4 subgroup, 139 Theatetus, 4 basic, 521 Third Isomorphism Theorem center, 155 groups, 165 cyclic, 141 modules, 298 generated by X, 143 Thompson, J. G., 219 index, 147 top element, 670 nontrivial, 139 topological group, 678 normal, 153 topological group, 461 generated by X, 158 topological space proper, 139 metric space, 673 pure, 364 topology, 675 subnormal, 192 p-adic, 675 torsion, 359 base, 675 submatrix, 581 compact, 674 submodule, 295 discrete, 678 cyclic, 296 finite index, 675 generated by X, 296 generated by S, 675 proper, 295 Hausdorff, 676 torsion, 379 induced, 676 subnormal subgroup, 192 Pr¨ufer, 676 subring, 32, 277 product, 678 subring generated by X, 280 subbase, 675 subspace, 249 torsion invariant, 295 group, 359 proper, 249 module, 380 smallest, 250 subgroup, 359 spanned by X, 250 submodule, 379 superalgebra, 572 torsion-free, 359, 380 support, 323 trace, 222 surjective, 238 Trace Theorem, 222 Suslin, A. A., 487 transcendence basis, 349 Sylvester, J. J., 426 transcendence degree, 351 symmetric transcendental element, 79 algebra, 559 transcendental extension, 353 bilinear form, 417 transfinite induction, 345 function, 208 transformation of direct system, 662 group, 117 transition matrix, 264 space, 417 transitive relation, 243 symmetric difference, 33, 129 transpose, 248 symmetric functions transposition, 117 elementary, 84, 180 twin primes, 16 symmetric group, 128, 242 type (pure extension field), 187 symmetric relation, 243 symmetry, 135 UFD, 105 symplectic Ulm, H., 372 706 Index

unique factorization domain, 105 unique factorization, k[x], 71 unit, 36 unitary group, 437 matrix, 437 transformation, 437 universal mapping problem, 449 solution, 449 upper bound, 210, 314

Vandermonde matrix, 589 Vandermonde, A.-T., 589 variety, 594 affine, 594 irreducible, 614 vector space, 247 Vi`ete, F., 3, 6

Watts, C. E., 663 wedge of p factors, 562 Weierstrass, K., 347 weight, 630 well-defined, 237 well-ordered, 316 Widman, J., 3 Wiles, A. J., 441, 593 Williams, K. S., 102 Wilson’s Theorem, 149 Wilson, J., 149

Yoneda, N., 528

Zariski closure, 602 topology on kn, 596 on Spec(R), 615 Zariski, O., 596 Zassenhaus Lemma, 195 modules, 302 Zassenhaus, H., 195 Zermelo, E. E. F., 442 zero divisor, 34 zero object, 459 zero of polynomial, 593 zero polynomial, 42 zero ring, 31 zero-divisor, 288 ZFC, 442 Zorn’s Lemma, 314 Zorn, M., 314 This new edition, now in two parts, has been significantly reor- ganized and many sections have been rewritten. This first part, designed for a first year of graduate algebra, consists of two courses: Galois theory and Module theory. Topics covered in the first course are classical formulas for solutions of cubic and quartic equations, classical number theory, commutative algebra, groups, and Galois theory. Topics in the second course are Zorn’s lemma, canonical forms, inner product spaces, categories and limits, tensor products, projective, injective, and flat modules, multilinear algebra, affine varieties, and Gröbner bases.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-165

GSM/165 AMS on the Web www.ams.orgwww.ams.org