CHAPTER 0
Introduction
The firs t noncommutativ e divisio n algebr a wa s define d b y Hamilto n aroun d
150 year s ago . Sinc e then , thes e object s hav e bee n a sourc e o f fascinatio n fo r
mathematicians o f many sorts. A division algebr a i s a very elementary object . I t i s
just a vector spac e wit h a n associativ e produc t structur e wher e d~
l
alway s exist s
for d nonzero . Despit e thi s simplicity , th e stud y o f divisio n algebra s ha s com e
to involv e a larg e collectio n o f mathematica l tool s an d viewpoints . Fo r me , thi s
contrast betwee n simplicit y an d diversit y i s a larg e par t o f thei r attraction . On e
begins wit h th e simples t o f objects , an d on e find s connection s t o numbe r theory ,
Galois theory , algebrai c geometry , algebrai c grou p theory , algebrai c K theory , an d
more specificall y t o Galoi s cohomology , etal e cohomology , an d geometri c invarian t
theory (an d this list is not exhaustive). O f course, the study of division algebras ha s
also influenced thes e same multiple areas of mathematics an d this cross fertilizatio n
is only growing .
The goa l o f thes e lecture s i s t o giv e a n introductio n t o th e theor y o f finite
dimensional divisio n algebra s tha t i s faithfu l t o thi s diversit y an d depth . O n th e
other hand , on e ha s t o kee p note s lik e thes e readable , s o one i s limited i n ho w fa r
one ca n explore . Th e reade r wil l find thi s tensio n throughou t th e notes .
In order not t o hide the diversity of the subject, I make no pretensions that thi s
material i s seif contained . I feel tha t t o d o so would inevitabl y distor t th e subject .
Instead I promis e t o b e honest . I giv e man y references , an d I hop e t o mak e clea r
which omitte d argument s ar e exercises , whic h ar e elementar y bu t ver y hard , an d
which requir e machiner y w e d o no t cove r o r eve n mention . O n th e othe r hand ,
one canno t reall y d o a grea t dea l i n finite space . Fo r thi s reason , I will , o n severa l
occasions, explor e th e beginning s o f a topic , an d the n sto p wit h wha t I hop e ar e
sufficient reference s fo r th e intereste d reade r t o g o on.
Of course , thes e note s reflec t m y individua l choices an d canno t hop e t o b e
anything lik e exhaustive . Ther e ar e man y man y topic s withi n thi s field, som e o f
which ar e never mentione d a t all . Other s ar e very briefly mentioned , bu t no t give n
the treatmen t the y deserve . I n makin g choices , I wa s particularl y intereste d i n
touching on those subjects wher e I feit I have something slightl y new to contribute ,
and wher e o f course I feel th e subjec t i s in som e sens e "abou t divisio n algebras. "
Though traditionall y a very importan t par t o f the theor y o f divisio n algebras ,
absolutely no t hing i s sai d i n thes e note s abou t involutions . Thi s despit e th e fac t
that divisio n algebras with involution provide a rieh set of connections to the theor y
of algebraic groups. However , I could never hope to add anything to the magnificen t
"Book o f Involutions" b y Knus , Merkurjev , Rost , an d Tignol , [K-T] .
I als o sa y very littl e abou t th e importan t are a o f divisio n algebra s wit h valua -
tions, particularly valuation s which ar e not discret e of rank 1. Thes e are importan t
methods, an d hav e yielde d importan t achievements . Her e m y choie e wa s base d o n
i
http://dx.doi.org/10.1090/cbms/094/01
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