Chapter 1 Finite Field s 1. Introductio n A field i s a n algebrai c structur e consistin g o f a se t o f element s fo r which the operation s o f addition, subtraction , multiplication , an d di - vision satisf y certai n prescribe d properties . Th e rea l number s ar e probably th e bes t know n example , alon g wit h th e fields o f rationa l numbers an d comple x numbers . Thes e ar e al l example s o f infinit e fields becaus e eac h contain s a n infinit e numbe r o f distinc t elements . Certain finite set s als o satisf y th e field propertie s whe n assigne d ap - propriate operations thes e finite fields ar e ou r focu s o f study i n thi s chapter. Because som e reader s ma y no t b e familia r wit h algebrai c struc - tures such as groups, rings, fields, and vector spaces, we have included a brie f introductio n t o the m i n Appendi x A . Th e materia l ther e in - cludes th e basi c definition s an d backgroun d theorem s require d here . 2. Finit e fields We begin ou r exploratio n o f finite fields b y determinin g th e possibl e sizes of a finite field. We will see that linea r algebra plays a crucial role in th e answe r t o thi s question . Recal l tha t ever y field ha s a uniqu e smallest subfield , calle d th e prime subfield, whic h i s the intersectio n of all of its subfield s (se e Exercise A.20) . 1
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