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.

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