PREFACE
This boo k aros e out o f a course entitle d
Mathematics in ancient and modern times:
The birth of modern algebra
which I gav e t o a smal l grou p o f freshme n student s i n UCL A i n th e Fal l o f 1993.
The student s ha d onl y hig h schoo l mathematic s a s thei r backgroun d an d s o I di d
not us e calculus . Howeve r I assisted the m s o that the y selecte d individua l project s
to wor k o n a s thei r assignmen t fo r fulfillin g th e requirement s o f th e course . Eac h
project wa s designe d t o hav e two parts : a historica l an d a mathematica l one . Th e
projects ha d wid e variety whic h surprise d me , bu t whic h increase d m y pleasur e a s
the cours e progressed . Thi s wa s th e firs t tim e tha t I ha d taugh t a smal l grou p o f
freshmen i n a forma t differen t fro m th e usua l cook-boo k typ e i n mor e tha n thirt y
years o f teaching i n UCLA , an d th e experienc e wa s memorable .
My inspiratio n fo r designin g an d teachin g suc h a cours e cam e afte r readin g
the beautifu l boo k Tales of physicists and mathematicians b y m y frien d Simo n
Gindikin, especiall y hi s accoun t o f th e lif e an d wor k o f Cardan o an d it s influenc e
on th e ris e o f moder n algebra . Howeve r i n th e cours e I di d no t confin e mysel f t o
the wor k o f Cardan o bu t explore d a wide r theme , namel y th e growt h o f algebr a
from it s ancien t origin s t o it s curren t stat e tha t allow s u s t o hav e a marvellou s
understanding o f th e whol e gamu t o f physica l sciences : fro m th e abstrac t reache s
of number theor y t o th e wonder s o f quantum physics .
In 1996 Rajendra Bhati a of the Indian Statistica l Institute suggested th e possi-
bility of making a book out o f these notes. I then revised the notes adding exercise s
and additiona l material—partl y historica l an d partl y dealin g wit h mor e advance d
topics— i n th e for m o f note s an d exercise s a t th e en d o f th e individua l chapters .
These ma y als o be use d a s starting point s fo r project s fo r th e mor e ambitiou s stu -
dent. Th e result is what I hope is an interesting way to present th e basics of modern
algebra a t a ver y elementar y level . Th e approac h tha t I hav e adopte d her e (a s i n
the course ) i s what Shafarevic h call s th e biogeneti c one ; th e studen t i s exposed t o
the historica l an d evolutionar y developmen t o f th e subjec t a s a n essentia l par t o f
the course , an d th e materia l i s presente d i n a wa y tha t allow s hi m o r he r t o se e
clearly ho w ideas hav e emerge d i n th e attemp t t o solv e specific questions .
I a m no t sur e ho w successfu l m y attemp t ha s been . Howeve r i t wa s a grea t
experience for me, not only at the time the course was given, but als o later when th e
book wa s written. I would lik e to recor d m y gratefulnes s t o th e Honor s Collegiu m
of UCLA which encouraged m e to give the course, to the small group of young me n
and wome n wh o too k th e origina l course , an d t o R . Bhati a wh o encourage d m e
XI
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