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 th e 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 stud y i n thi s
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 include d
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 begi n ou r exploratio n o f finite fields b y determinin g th e possibl e
sizes of a finite field. W e 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 th e intersectio n
of al l o f it s subfield s (se e Exercis e A.20) .
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