http://dx.doi.org/10.1090/fim/021 FIELDS INSTITUT E MONOGRAPHS THE FIELD S INSTITUT E FO R RESEARC H I N MATHEMATICA L SCIENCE S Brauer Typ e Embedding Problems Arne Lede t American Mathematica l Societ y Providence, Rhode Islan d The Field s Institut e for Researc h i n Mathematical Science s The Field s Institute i s a center fo r mathematica l researc h activity , locate d i n Toronto , Canada. Ou r missio n i s to provid e a supportiv e an d stimulatin g environmen t fo r math - ematics research , innovatio n an d education . Th e Institut e i s supporte d b y th e Ontari o Ministry o f Training, College s and Universities , the Natural Science s and Engineerin g Re - search Counci l o f Canada , an d seve n Ontari o universitie s (Carleton , McMaster , Ottawa , Toronto, Waterloo , Wester n Ontario , an d York) . I n additio n ther e ar e severa l affiliate d universities an d corporat e sponsor s i n both Canad a an d th e Unite d States . Fields Institut e Editoria l Board : Car l R . Rieh m (Managin g Editor) , Barbar a Le e Keyfitz (Directo r o f the Institute) , Thoma s S . Salisbur y (Deput y Director) , Joh n Blan d (Toronto), Kennet h R . Davidso n (Waterloo) , Joe l Feldma n (UBC) , R . Mar k Goresk y (Institute fo r Advance d Study , Princeton) , Camero n Stewar t (Waterloo) , Norik o Yu i (Queen's). The autho r wa s supporte d i n par t b y a Queen' s Universit y Advisor y Researc h Com - mittee Postdoctora l Fellowship . 2000 Mathematics Subject Classification. Primar y 12F12 , 16K50 ; Secondar y 16S35 , 12G05. For additiona l informatio n an d update s o n this book , visi t www.ams.org/bookpages/fim-21 Library o f Congres s Cataloging-in-Publicatio n Dat a Ledet, Arne , 1967 - Brauer typ e embeddin g problem s / Arn e Ledet . p. cm . (Field s Institut e monographs , ISS N 1069-5273 ; 21 ) Includes bibliographica l reference s an d index . ISBN 0-8218-3726- 5 (alk . paper ) 1. Inverse Galoi s theory . 2 . Brauer groups . I . Title . II . Series . QA247 + 200504281 3 Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitted t o mak e fai r us e o f the material , suc h a s to cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f the sourc e i s given . Republication, systemati c copying , o r multiple reproduction o f any material i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Acquisition s Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Request s ca n als o b e mad e b y e-mail t o [email protected] . © 200 5 b y the America n Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o th e Unite d State s Government . Printed i n the Unite d State s o f America . @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . This publicatio n wa s prepare d b y The Field s Institute . Visit th e AM S hom e pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 2 1 1 0 09 0 8 0 7 0 6 0 5 Contents Introduction v Acknowledgments vii i Chapter 1 . Galoi s Theor y 1 Notation 1 1.1. Integra l element s 1 1.2. Algebrai c extension s an d splittin g fields 2 1.3. Separabilit y 4 1.4. Galoi s extensions 6 1.5. Th e Galoi s group a s permutation grou p 9 1.6. Reductio n modul o prime s 1 1 1.7. Cyclotomi c fields 1 3 1.8. Cycli c extensions an d Hilber t Sat z 9 0 1 5 1.9. Radica l extension s an d solvabilit y 1 7 1.10. Th e norma l basi s theore m 1 9 Exercises 2 1 Chapter 2 . Invers e Galoi s Theor y an d Embeddin g Problem s 2 7 2.1. Invers e Galoi s theory 2 7 2.2. Definitio n o f Galoi s theoretical embeddin g problem s 2 9 2.3. Grou p cohomolog y 3 0 2.4. Braue r typ e embeddin g problem s 3 4 Exercises 4 0 Chapter 3 . Braue r Group s 4 3 3.1. Basi c fact s abou t algebra s 4 3 3.2. Simpl e Artinia n ring s 4 4 3.3. Finite-dimensiona l centra l simpl e algebra s 4 6 3.4. Th e Braue r grou p 5 0 3.5. Cycli c algebras 5 3 3.6. Braue r typ e embeddin g problem s revisite d 5 6 3.7. Extension s o f the dihedra l grou p 5 9 Exercises 6 2 Chapter 4 . Grou p Cohomolog y 6 7 4.1. Th e cohomologie s i n dimension s 0 and 1 6 7 4.2. Galoi s Twis t 6 8 4.3. Th e long-exac t cohomolog y sequenc e 7 0 4.4. Th e corestrictio n ma p 7 2 iv Content s Exercises 7 3 Chapter 5 . Quadrati c Form s 7 7 5.1. Basi c theory o f quadratic form s 7 7 5.2. Cliffor d algebra s 8 1 5.3. Quadrati c form s an d non-abelia n cohomolog y 8 3 Exercises 8 5 Chapter 6 . Decomposin g th e Obstructio n 8 7 6.1. Decompositio n alon g a direct produc t 8 7 6.2. Orthogona l representation s 9 5 6.3. Projectiv e representation s 9 8 Exercises 10 2 Chapter 7 . Quadrati c Form s an d Embeddin g Problem s 10 5 7.1. Group s o f exponent 4 10 5 7.2. Witt' s Criterio n 10 8 7.3. Group s o f exponent 8 11 4 Exercises 12 1 Chapter 8 . Reducin g th e Embeddin g Proble m 12 3 8.1. Reductio n t o Brauer typ e embeddin g problem s 12 3 8.2. Examples : Th e holomorp h o f the quaternio n grou p 13 3 8.3. Embeddin g problem s wit h cycli c kernel o f order 4 14 1 Exercises 14 8 Appendix A . Pro-finit e Galoi s Theor y 15 1 Introduction 15 1 A.l. Th e separabl e closur e o f a field 15 1 A.2. Galoi s extension s 15 2 A.3. Pro-finit e group s 15 4 A.4. Pro-finit e Galoi s theory 15 6 A.5. Pro-finit e grou p cohomolog y 15 7 A.6. Pro-finit e cohomolog y an d th e Braue r grou p 15 8 Exercises 16 0 Bibliography 16 3 Index 167 Introduction Consider a Galoi s theoretica l embeddin g problem , i.e. , th e questio n o f embeddin g a Galoi s extensio n M/K wit h Galoi s grou p G — Gal(M/K) int o a large r Galoi s extension F/K, suc h tha t th e Galoi s grou p Gal(F/K) i s isomorphi c t o a specifie d group E an d th e restrictio n ma p fro m Ga l (F/if) t o G correspond s t o a give n homomorphism n: E — > G. Ho w d o w e approac h suc h a problem ? Ho w d o w e determine whethe r suc h a n extensio n F/K exists ? Ho w d o w e find i t i f it does ? O r (preferably) ho w d o w e find al l o f them i f there ar e any ? The answer s t o thes e question s o f cours e depen d o n th e kin d o f embeddin g problem considered , bot h with respec t t o the nature o f the groups an d the nature o f the field. Fo r instance, i f G and E ar e cyclic , it takes nothing mor e than elementar y Galois theor y t o solv e th e proble m ove r a finite field, wherea s i t take s clas s field theory (o r a t leas t som e reasonably sophisticate d algebrai c numbe r theory ) t o solv e it ove r th e field o f rational numbers .