Preface The ai m o f this boo k i s to provide a brief 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 been greatl y expande d int o the 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 of properties o f finite fields. Man y of these propertie s are used in later chapters 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 of finite fields and 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. vn
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