By Robert J. McEliece

This booklet built from a path on finite fields I gave on the collage of Illinois at Urbana-Champaign within the Spring semester of 1979. The path used to be taught on the request of an excellent team of graduate scholars (includ ing Anselm Blumer, Fred Garber, Evaggelos Geraniotis, Jim Lehnert, Wayne Stark, and Mark Wallace) who had simply taken a path on coding thought from me. the speculation of finite fields is the mathematical origin of algebraic coding concept, yet in coding thought classes there's by no means a lot time to offer greater than a "Volkswagen" therapy of them. yet my 1979 scholars sought after a "Cadillac" therapy, and this e-book differs little or no from the path I gave in reaction. seeing that 1979 i've got used a subset of my direction notes (correspond ing approximately to Chapters 1-6) because the textual content for my "Volkswagen" remedy of finite fields at any time when I educate coding thought. there's, paradoxically, no coding thought anyplace within the booklet! If this booklet had an extended name it might be "Finite fields, ordinarily of char acteristic 2, for engineering and computing device technological know-how functions. " It definitely doesn't fake to hide the overall idea of finite fields within the profound intensity that the new publication of Lidl and Neidereitter (see the Bibliography) does.

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Now for each positive divisor t of q - 1, denote by 1/J(t) the number of elements of order t in F. 6) L tlq-l 1/J(t) = q - 1. 5) we see that L (r/>(t) -1/J(t)) = o. 5, r/>(t) -1/J(t) t/q-1. ~ 0, for all t. Thus r/>(t) = 1/J(t) for all • Corollary. In every finite field, there exists at least one element (in fact, exactly r/>(q - 1) elements) of order q - 1. Hence, the multiplicative group of any finite field is cyclic. Definition. , a generator of the cyclic group F* = F - {O}, is called a primitive root of the field F.

We come now to a new topic, minimal polynomials. 1 F has pm elements for some prime p and some positive integer m. 1 we saw that F could be viewed as an m-dimensional vector space over Fp. Now let 0:' be an arbitrary element of F. Consider the m + 1 elements of F. • , Am, not all zero, such that In other words, if A(x) = Ao + A1x + ... 7) A(x) = O. Of course, 0:' may satisfy other polynomial equations, and so we define 8(0:') to be the set of all such polynomials: 8(0:') = {f(x) E Fp(x) : /(0:') = O}.

Indeed it is isomorphic to the field Fp = Z mod p = {O, 1, ... ,p - 1}, if we make the obvious identification Ui i. Thus it is possible to view F as a vector space over Fp. W2, ••. 3) where each ai is an element of Fp. 3) that the field contains exactly pm elements. 1: • The last part of the proof, where we invoked a little of the theory of vector spaces, can be done directly, as follows: Abstract Properties of Finite Fields 31 "We have Fp as a subfield of F. Let Wl E F - Fp; there are p elements in F of the form alWt.