Finite Fields for Computer Scientists and Engineers by Robert J. McEliece

By Robert J. McEliece

This booklet built from a direction on finite fields I gave on the college of Illinois at Urbana-Champaign within the Spring semester of 1979. The path used to be taught on the request of a great crew of graduate scholars (includ­ ing Anselm Blumer, Fred Garber, Evaggelos Geraniotis, Jim Lehnert, Wayne Stark, and Mark Wallace) who had simply taken a direction on coding concept from me. the idea of finite fields is the mathematical origin of algebraic coding idea, yet in coding concept classes there's by no means a lot time to provide greater than a "Volkswagen" therapy of them. yet my 1979 scholars sought after a "Cadillac" remedy, and this e-book differs little or no from the direction I gave in reaction. considering that 1979 i've got used a subset of my path notes (correspond­ ing approximately to Chapters 1-6) because the textual content for my "Volkswagen" therapy of finite fields each time I train coding idea. there's, paradoxically, no coding idea wherever within the ebook! If this booklet had an extended name it'd be "Finite fields, commonly of char­ acteristic 2, for engineering and computing device technology purposes. " It definitely doesn't fake to hide the final thought of finite fields within the profound intensity that the hot booklet of Lidl and Neidereitter (see the Bibliography) does.

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X - a q d-I ) = Adx d + Ad-Ixd - I + ... 16) each coefficient Ai is an element of F. 12. 12, (x - (3)q = xq + (-l)q{3q = x q - {3q. 19) are equal, we must have A1 = Ai, i = 0, 1, ... 10) Ai E k for all i. 15), is a divisor of the minimal polynomial of a, and that the coefficients of la(x) all lie in the small field k. 9, we conclude that la(x) must be the minimal polynomial of a. We state this important conclusion in a theorem. 14. Let F be a field with qn elements, and k be a q-element subJield.

Repeats with period 2. Show that in general, the period of the decimal representation of the fraction kip, where p is an odd prime (except 5) and 1 $ k $ p - 1, is equal to the order of 10 in the field F". Why is 5 exceptional? Chapter 6 Finite Fields Exist and are Unique Let us recapitulate our results so far. We know that there is, for every prime p, a finite field with p elements, viz. the integers (mod p). This field we have denoted by Fp. 1 that provided there is an irreducible polynomial of degree mover Fp , there exists a field with pm elements.

We can compute the orders of o:i, i = 0, 1, ... 4; the work is summarized in the following table: i gcd(i,12) ord(o:i) 0 1 2 3 12 1 2 3 4 1 6 1 4 3 2 1 1 12 6 4 3 12 2 12 3 4 6 12 4 5 6 7 8 9 10 11 This is a bit surprising. Given only the fact that there exists at least one element of order 12 in F, it follows that there are exactly 4 elements of order 12! Furthermore, there are exactly 2 elements of order 6, 2 of order 4, 2 of order 3, 1 of order 2, and 1 of order 1. Notice that to draw these conclusions, we only had do calculations with integers, and didn't need to know anything about F.

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