An Introduction to Central Simple Algebras and Their Applications to Wireless Communication

Mathematical Surveys and Monographs Volume 191

Grégory Berhuy Frédérique Oggier

American Mathematical Society http://dx.doi.org/10.1090/surv/191

An Introduction to Central Simple Algebras and Their Applications to Wireless Communication

Mathematical Surveys and Monographs Volume 191

An Introduction to Central Simple Algebras and Their Applications to Wireless Communication

Grégory Berhuy Frédérique Oggier

American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Ralph L. Cohen, Chair Benjamin Sudakov Robert Guralnick MichaelI.Weinstein MichaelA.Singer

2010 Mathematics Subject Classiﬁcation. Primary 12E15; Secondary 11T71, 16W10.

For additional information and updates on this book, visit www.ams.org/bookpages/surv-191

Library of Congress Cataloging-in-Publication Data Berhuy, Grégory. An introduction to central simple algebras and their applications to wireless communications /Grégory Berhuy, Frédérique Oggier. pages cm. – (Mathematical surveys and monographs ; volume 191) Includes bibliographical references and index. ISBN 978-0-8218-4937-8 (alk. paper) 1. Division algebras. 2. Skew fields. I. Oggier, Frédérique. II. Title.

QA247.45.B47 2013 512.3–dc23 2013009629

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 a chapter 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. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294 USA. Requests can also be made by e-mail to [email protected]. c 2013 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 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 181716151413 Contents

Foreword...... vii

Introduction...... 1

ChapterI. Centralsimplealgebras...... 3 I.1. Preliminaries on k-algebras...... 3 I.2. Centralsimplealgebras: thebasics...... 7 I.3. Introducingspace-timecoding...... 11 Exercises...... 18

ChapterII. Quaternionalgebras...... 21 II.1. Propertiesofquaternionalgebras...... 21 II.2. Hamiltonquaternions...... 27 II.3. Quaternionalgebrasbasedcodes...... 28 Exercises...... 30

Chapter III. Fundamental results on central simple algebras...... 31 III.1. Operationsoncentralsimplealgebras...... 31 III.2. Simplemodules...... 35 III.3. Skolem-Noether’stheorem...... 43 III.4. Wedderburn’stheorem...... 45 III.5. Thecentralizertheorem...... 47 Exercises...... 50

ChapterIV. Splittingﬁeldsofcentralsimplealgebras...... 53 IV.1. Splittingﬁelds...... 53 IV.2. Thereducedcharacteristicpolynomial...... 60 IV.3. Theminimumdeterminantofa code...... 68 Exercises...... 76

ChapterV. TheBrauergroupofa ﬁeld...... 79 V.1. DeﬁnitionoftheBrauergroup...... 79 V.2. Brauerequivalenceandbimodules...... 82 V.3. Indexandexponent...... 91 Exercises...... 98

ChapterVI. Crossedproducts...... 101 VI.1. Deﬁnitionofcrossedproducts...... 101 VI.2. Somepropertiesofcrossedproducts...... 108 VI.3. Shapingandcrossedproductsbasedcodes...... 118 Exercises...... 126

v vi CONTENTS

ChapterVII. Cyclicalgebras...... 129 VII.1. Cyclicalgebras...... 129 VII.2. Centralsimplealgebrasoverlocalfields...... 137 VII.3. Centralsimplealgebrasovernumberfields...... 139 VII.4. Cyclicalgebrasofprimedegreeovernumberfields...... 141 VII.5. Examples...... 144 VII.6. Cyclicalgebrasandperfectcodes...... 150 VII.7. Optimalityofsomeperfectcodes...... 156 Exercises...... 163 ChapterVIII. Centralsimplealgebrasofdegree4...... 165 VIII.1. A theoremofAlbert...... 165 VIII.2. Structureofcentralsimplealgebrasofdegree4...... 168 VIII.3. Albert’sTheorem...... 176 VIII.4. Codesoverbiquadraticcrossedproducts...... 178 Exercises...... 187 ChapterIX. Centralsimplealgebraswithunitaryinvolutions...... 189 IX.1. Basicconcepts...... 189 IX.2. Thecorestrictionalgebra...... 191 IX.3. Existenceofunitaryinvolutions...... 198 IX.4. Unitaryinvolutionsoncrossedproducts...... 203 IX.5. Unitaryspace-timecoding...... 209 Exercises...... 228 AppendixA. Tensorproducts...... 231 A.1. Tensorproductofvectorspaces...... 231 A.2. Basicpropertiesofthetensorproduct...... 235 A.3. Tensor product of k-algebras...... 242 AppendixB. A glimpseofnumbertheory...... 249 B.1. Absolutevalues...... 249 B.2. Factorizationofidealsinnumberfields...... 253 B.3. Absolutevaluesonnumberfieldsandcompletion...... 262 AppendixC. Complexideallattices...... 265 C.1. Generalitiesonhermitianlattices...... 265 C.2. Complexideallattices...... 266

Bibliography...... 271 Index...... 275 Foreword

Mathematics continually surprises and delights us with how useful its most abstract branches turn out to be in the real world. Indeed, physicist Eugene Wigner’s mem- orable phrase1 “The unreasonable effectiveness of mathematics” captures a critical aspect of this utility. Abstract mathematical ideas often prove to be useful in rather “unreasonable” situations: places where one, a priori, would not expect them at all! For instance, no one who was not actually following the theoretical explorations in multi-antenna wireless communication of the late 1990s would have predicted that division algebras would turn out to be vital in the deployment of multi-antenna communication. Yet, once performance criteria for space-time codes (as coding schemes for multi-antenna environments are called) were developed and phrased as a problem of design of matrices, it was completely natural that division algebras should arise as a solution of the design problem. The fundamental performance criterion ask for n × n matrices Mi such that the difference of any two of the Mi is of full rank. To anyone who has worked with division algebras, the solution simply leaps out: any division algebra of index n embeds into the n × n matrices over a suitable field, and the matrices arising from the embedding naturally satisfy this criterion. But there is more. Not only did division algebras turn out to be the most natural context in which to solve the fundamental design problem above, they also proved to be the correct objects to satisfy various other performance criteria that were developed. For instance, a second, and critical, performance criterion called the coding gain criterion turned out to be naturally satisfied by considering division algebras over number fields and using natural Z-orders within them that arise from rings of integers of maximal subfields. Other criteria (for instance “DMG optimality,” “good shaping,” “information-losslessness” to name just a few) all turned out to be satisfied by considering suitable orders inside suitable division algebras over number fields. Indeed, this exemplifies another phenomenon Wigner describes: after saying that “mathematical concepts turn up in entirely unexpected connections,” he goes on to say that “they often permit an unexpectedly close and accurate description of the phenomena in these connections.” The match between division algebras and the requirement of space-time codes is simply uncanny. The subject of multi-antenna communication has several unsolved mathematical problems still, for instance, in the area of decoding for large numbers of antennas. Nevertheless, division algebras are already being deployed for practical two-antenna

1Eugene P. Wigner, The unreasonable eﬀectiveness of mathematics in the natural sciences, Comm. Pure Appl. Math., 13 Feb. 1960, 1–14

vii viii FOREWORD systems, and codes based on them are now part of various standards of the Insti- tute of Electrical and Electronics Engineers (IEEE). It would behoove a student of mathematics, therefore, to know something about the applicability of division algebras while studying their theory; in parallel, it is vital for a communications engineer working in coding for multiple-antenna wireless to know something about division algebras. Berhuy and Oggier have written a charming text on division algebras and their ap- plication to multiple-antenna wireless communication. There is a wealth of examples here, particularly over number fields and local fields, with explicit calculations, that one does not see in other texts on the subject. By pairing almost every chapter with a discussion of issues from wireless communication, the authors have given a very concrete flavor to the subject of division algebras. The book can be studied profitably not just by a graduate student in mathematics, but also by a mathe- matically sophisticated coding theorist. I suspect therefore that this book will find wide acceptability in both the mathematics and the space-time coding community and will help cross-communication between the two. I applaud the authors’ efforts behind this very enjoyable book. B.A. Sethuraman Northridge, California Bibliography

1. S.M. Alamouti, A simple transmit diversity technique for wireless communications, IEEE J. Selected Areas Communications 16 (1998), 1451–1458. 2. A.A. Albert, Structure of Algebras, A.M.S. Coll.Pub., vol. 24, AMS, 1939. 3. S.A. Amitsur, On central division algebras,IsraelJ.Math.12 (1972), no. 4, 408–420. 4. E. Bayer-Fluckiger, F. Oggier, and E. Viterbo, New algebraic constructions of rotated Zn–lattice constellations for the Rayleigh fading channel, IEEE Transactions on Information Theory 50 (2004), no. 4. 5. J.-C. Belfiore and G. Rekaya, Quaternionic lattices for space-time coding, Proceedings of IEEE Information Theory Workshop (ITW), Paris (2003). 6. J.-C. Belfiore, G. Rekaya, and E. Viterbo, The Golden code: A 2 × 2 full rate space-time code with non vanishing determinants, IEEE Trans. on Inf. Theory 51 (2005), no. 4. 7. G. Berhuy, Algebraic space-time codes based on division algebras with a unitary involution, (2012), Preprint. Available from http://www-fourier.ujf-grenoble.fr/~berhuy/fichiers/ unitarycodes.pdf. 8. G. Berhuy and F. Oggier, Space-time codes from crossed product algebras of degree 4,Pro- ceedings of Applied algebra, algebraic algorithms and error-correcting codes, Lecture Notes in Comput. Sci 4851 (2007), 90–99. 9. , On the existence of perfect space-time codes, IEEE Transactions on Information Theory 55 (2009), no. 5, 2078–2082. 10. G. Berhuy and R. Slessor, Optimality of codes based on crossed product algebras, (2011), Preprint. Available from http://www-fourier.ujf-grenoble.fr/~berhuy/ Berhuy-Slessor-2209.pdf. 11. J.W.S. Cassels and A. Frölich, Algebraic number theory, Acad. Press, 1967. 12. M. O. Damen, A. Tewfik, and J.-C. Belfiore, A construction of a space-time code based on number theory, IEEE Transactions on Information Theory 48 (2002), no. 3. 13. P. Elia, B.A. Sethuraman, and P. V. Kumar, Perfect space-time codes for any number of antennas, IEEE Transactions on Information Theory 53 (2007). 14. Jr. G. Forney, R. Gallager, G. Lang, F. Longstaff, and S. Qureshi, Efficient modulation for band-limited channels, IEEE Journal on Selected Areas in Communications 2 (1984), no. 5, 632–647. 15. A. Fröhlich and M. Taylor, Algebraic number theory, Cambridge University Press, 1991. 16. P. Gille and T. Szamuely, Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics, vol. 101, Cambridge University Press, 2006. 17. D. Haile, A useful proposition for division algebras of small degree, Proc. Amer. Math. Soc. 106 (1989), no. 2, 317–319. 18. T. Hanke, A twisted Laurent series ring that is a noncrossed product, Israel J. Math. 150 (2005). 19. B. Hassibi and B. Hochwald, Cayley differential unitary space-time codes, IEEE Trans. on Information Theory 48 (2002). 20. J. Hiltunen, C. Hollanti, and J. Lahtonen, Dense full-diversity matrix lattices for four transmit antenna MISO channel, Proceedings of IEEE International Symposium on Information Theory (ISIT) (2005). 21. B. Hochwald and W. Sweldens, Differential unitary space time modulation, IEEE Trans. Commun. 48 (2000). 22. J. G. Huard, B. K. Spearman, and K. S. Williams, Integral bases for quartic fields with quadratic subfields,J.NumberTheory51 (1995), no. 1, 87–102. MR1321725 (96a:11115) 23. B. Hughes, Differential space-time modulation, IEEE Trans. Inform. Theory 46 (2000).

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24. K. Iyanaga, Class numbers of definite hermitian forms, J. Math. Soc. Japan 21 (1969). 25. N. Jacobson, Finite-dimensional division algebras over fields, Springer-Verlag, Berlin, 1996. 26. G. J. Janusz, Algebraic number fields, second ed., Graduate Studies in Mathematics, vol. 7, American Mathematical Society, Providence, RI, 1996. 27. S. Karmakar and B Sundar Rajan, High-rate, multi-symbol-decodable STBCs from Clifford algebras, IEEE Transactions on Information Theory 55 (2009), no. 6. 28. A. W. Knapp, Advanced algebra, Cornerstones, Birkhäuser Boston Inc., Boston, MA, 2007, Along with a companion volume ıt Basic algebra. 29. M.A. Knus, A. Merkurjev, M. Rost, and J.-P. Tignol, The book of involutions,Amer.Math. Soc. Coll. Pub., vol. 44, A.M.S, 1998. 30. H. Koch, Algebraic number theory, Springer-Verlag,Berlin, 1997. 31. , Number theory: algebraic numbers and functions, Graduate Studies in Mathematics, vol. 24, A.M.S, 2000. 32. J. Lahtonen, N. Markin, and G. McGuire, Construction of space-time codes from division algebras with roots of unity as non-norm elements, IEEE Trans. on Information Theory 54 (2008), no. 11. 33. F. Oggier, On the optimality of the Golden code, IEEE Information Theory Workshop (ITW’06) (2006). 34. , Cyclic algebras for noncoherent differential space-time coding, IEEE Transactions on Information Theory 53 (9) (2007), 3053–3065. 35. , A survey of algebraic unitary codes, International Workshop on Coding and Cryp- tology 5757 (2009), 171–187. 36. F. Oggier, J.-C. Belfiore, and Viterbo E., Cyclic division algebras: A tool for space-time coding, Now Publishers Inc., Hanover, MA, USA, 2007. 37. F. Oggier and B. Hassibi, “an algebraic family of distributed space-time codes for wireless relay networks, Proceedings of IEEE International Symposium on Information Theory (ISIT) (2006). 38. , Algebraic cayley differential space-time codes, IEEE Transactions on Information Theory 53 (2007), no. 5. 39. F. Oggier and L. Lequeu, Families of unitary matrices achieving full diversity, International Symposium on Information Theory (2005), 1173–1177. 40. F. Oggier, G. Rekaya, J.-C. Belfiore, and E. Viterbo, Perfect space-time block codes, IEEE Trans. Inform. Theory 52 (2006), no. 9. 41. R.S. Pierce, Associative algebras, Graduate Texts in Mathematics, vol. 88, Springer-Verlag, New York-Berlin, 1982. 42. S. Pumpluen and T. Unger, Space-time block codes from nonassociative division algebras, Advances in Mathematics of Communications 5 (2011), no. 3. 43. P. Ribenboim, Classical theory of algebraic numbers, Universitext, Springer, 2001. 44. J.-P. Serre, Corps locaux (4ème édition), Hermann, 1997. 45. B. A. Sethuraman, Division algebras and wireless communication, Notices of the AMS 57 (2010), no. 11. 46. B.A. Sethuraman and B. Sundar Rajan, An algebraic description of orthogonal designs and the uniqueness of the Alamouti code, Proceedings of GlobCom (2002). 47. , STBC from field extensions of the rational field, Proceedings of IEEE International Symposium on Information Theory (ISIT), Lausanne (2002). 48. B.A. Sethuraman, B. Sundar Rajan, and V. Shashidhar, Full-diversity, high-rate space-time block codes from division algebras, IEEE Transactions on Information Theory 49 (2003). 49. A. Shokrollahi, B. Hassibi, B.M. Hochwald, and W. Sweldens, Representation theory for high- rate multiple-antenna code design, IEEE Trans. Information Theory 47 (2001), no. 6. 50. R. Slessor, Performance of codes based on crossed product algebras,Ph.D.thesis,School of Mathematics, Univ. of Southampton, May 2011 note = Available from http:// eprints.soton.ac.uk/197309/3.hasCoversheetVersion/Thesis_--_Richard_Slessor.pdf, OPTannote = annote, 2011. 51. H. P. F. Swinnerton-Dyer, A brief guide to algebraic number theory, Cambridge University Press, 2001. 52. V. Tarokh, N. Seshadri, and A. Calderbank, Space-time codes for high data rate wireless communication : Performance criterion and code construction,IEEETrans.Inform.Theory 44 (1998), 744–765. BIBLIOGRAPHY 273

53. The PARI Group, Bordeaux, PARI/GP, version 2.3.3, 2005, available from http://pari. math.u-bordeaux.fr/. 54. J.-P. Tignol, Produits crois´es ab´eliens, J.of Algebra 70 (1981). 55. T. Unger and N. Markin, Quadratic forms and space-time block codes from generalized quaternion and biquaternion algebras, IEEE Trans. on Information Theory 57 (2011), no. 9. 56. R. Vehkalahti, C. Hollanti, J. Lahtonen, and K. Ranto, On the densest MIMO lattices from cyclic division algebras, IEEE Transactions on Information Theory 55 (2009), no. 8. 57. S. Vummintala, B. Sundar Rajan, and B.A. Sethuraman, Information-lossless space-time block codes from crossed-product algebras, IEEE Trans. Inform. Theory 52 (2006), no. 9. 58. H. Yao and G.W. Wornell, Achieving the full MIMO diversity-multiplexing frontier with rotation-based space-time codes, Proceedings of Allerton Conf. on Communication, Control and Computing (2003).

Index

k-algebra degree, 46 center of a, 4 different ideal, 259 central, 8 differential modulation, 209 definition, 3 discriminant ideal, 259 morphism, 3 diversity, 14 quaternion, 8 simple, 7 elementary tensor, 233 split, 9 exponent, 95 absolute discriminant, 259 fading matrix, 11 absolute value Frobenius map, 253 p-adic, 262 fully diverse code, 14 archimedean, 249 Goldman element, 86 definition, 249 discrete, 250 Hasse symbol, 138 equivalence, 249 extension, 251 ideal (ramification) non-archimedean, 249 inert, 255 absolutevalue ramification index, 255 extension ramified, 255 totally ramified, 251 tamely ramified, 255 ramification index, 251 totally ramified, 255 ramified, 251 totally split, 255 residual degree, 251 unramified, 255 unramified, 251 wildly ramified, 255 index, 46 bimodule, 82 information symbol, 12 Brauer equivalence, 46 inner automorphism, 43 Brauer group, 81 involution relative, 82 definition of an, 189 of the first kind, 189 canonical involution, 228 of the second kind, 189 centralizer, 31 coboundary, 113 local parameter, 251 cocycle, 104 codebook, 13 MIMO, 11 coding gain, 14 module coherence interval, 12 definition, 35 coherent, 13 finitely generated, 36 cohomologous cocycles, 113 free, 37 corestriction, 195 morphism, 36 crossed product, 107 rank, 41 cyclic algebra, 130 non-coherent, 209 decomposition group, 261 norm of an ideal

275 276 INDEX

absolute norm, 258 relative norm, 258 number field, 253 opposite algebra, 34 place, 249 complex, 262 finite, 262 real, 262 prime ideals residual degree, 255 ramification groups, 261 rate, 14, 17 reduced characteristic polynomial, 63 reduced norm, 66 reduced trace, 66 residue field, 250 restriction map, 82 ring of integers, 253

Sandwich morphism, 35 semilinear map, 191 simple module, 39 SNR, 13 space-time codes, 13 splitting field, 53 subalgebra definition, 3 subfield, 9 submodule, 36 tensor product of algebras, 5, 243 of vector spaces, 231 trace form, 76 valuation ring, 250 Selected Published Titles in This Series

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For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/survseries/. Central simple algebras arise naturally in many areas of mathematics. They are gory Berhuy closely connected with ring theory, but are é also important in representation theory, algebraic geometry and number theory. Recently, surprising applications of the

theory of central simple algebras have Photograph courtesy of Gr A. Datta Photograph courtesy of arisen in the context of coding for wireless communication. The exposition in the book takes advantage of this serendipity, presenting an introduction to the theory of central simple algebras intertwined with its applications to coding theory. Many results or constructions from the standard theory are presented in classical form, but with a focus on explicit techniques and examples, often from coding theory. Topics covered include quaternion algebras, splitting fields, the Skolem-Noether Theorem, the Brauer group, crossed products, cyclic algebras and algebras with a unitary involution. Code constructions give the opportunity for many examples and explicit computations. This book provides an introduction to the theory of central algebras accessible to graduate students, while also presenting topics in coding theory for wireless communication for a mathematical audience. It is also suitable for coding theorists interested in learning how division algebras may be useful for coding in wireless communication.

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