The ai m o f this boo k i s to provid e a brie f introductio n t o finite field s
and som e o f thei r man y fascinatin g applications . Th e boo k aros e
from lecture s o f th e first autho r i n a cours e entitle d "Finit e Field s
and Thei r Applications, " whic h wa s taugh t i n th e Departmen t o f
Mathematics a t Th e Pennsylvani a Stat e Universit y durin g th e Fal l
semester o f 2004 . Th e cours e wa s par t o f th e department' s Math -
ematics Advance d Stud y Semester s (MASS) program . Th e secon d
author produce d a n initia l onlin e se t o f note s fro m thes e lectures ,
which hav e bee n greatl y expande d int o th e presen t volume .
The mos t importan t chapte r o f thi s tex t i s th e first, whic h dis -
cusses a variety o f properties o f finite fields. Man y o f these propertie s
are used in later chapter s where various applications of finite fields are
discussed. Th e chapte r begin s wit h a discussio n o f th e basi c proper -
ties o f finite fields an d extensio n fields. I t the n define s th e importan t
trace an d nor m function s an d establishe s som e o f thei r properties .
Bases fo r extensio n fields, includin g dual , normal , an d primitiv e nor -
mal bases , ar e the n discussed . Th e firs t chapte r conclude s wit h a
few result s concernin g polynomial s ove r finite fields. Thes e includ e
a discussio n o f th e orde r o f a polynomial , formula s fo r th e numbe r
and order s o f irreducibl e polynomials , an d propertie s o f linearize d
polynomials an d permutatio n polynomial s ove r finite fields.
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