By Robert J. McEliece

This ebook 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 was once taught on the request of an excellent crew 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 idea from me. the idea of finite fields is the mathematical beginning of algebraic coding thought, yet in coding thought classes there's by no means a lot time to offer greater than a "Volkswagen" remedy of them. yet my 1979 scholars sought after a "Cadillac" therapy, and this ebook differs little or no from the direction I gave in reaction. seeing 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 every time I educate coding conception. there's, paradoxically, no coding concept wherever within the ebook! If this e-book had an extended name it might be "Finite fields, normally of char acteristic 2, for engineering and computing device technological know-how purposes. " It definitely doesn't fake to hide the overall idea of finite fields within the profound intensity that the new booklet of Lidl and Neidereitter (see the Bibliography) does.

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**Example text**

2. Let D = Z, the integers, p = 13. Then D mod p has 13 elements, which we may denote by 0, I, ... , 12. Then for example Let us find the inverse of 6. We apply Euclid's algorithm to 6 and 13 to find a linear combination of 6 and 13 equal to 1. We find that 6 . 11 - 13 . 5 = 1. , (6)- = 11. 3. Take D = Z again, but now let p be an arbitrary prime. The resulting important finite field has exactly p elements {O, I, ... , p - l}j it is commonly denoted by either of the two symbols Fp or GF(p}. This construction yields infinitely many finite fields, since there are infinitely many • primes.

6 we know that this polynomial will factor uniquely into a product of irreducible monic polynomials over k. The next theorem tells us something more about this factorization. 1. xqn - X = II Vd(X), din where Vd(X) is the product of all monic irreducible polynomials in k[x] of degree d. Proof: Let d be a divisor of n and let f(x) be a monic irreducible polynomial of degree dover k. Form the field F = k[x] (modf(x))j then F has qd elements. , a = x. 10, a qd = a, and this is equivalent to the statement d f(x) I (x q - x).

Let a k = ak+t be the first repeat in the sequence. Then clearly k = OJ otherwise a k- l = ak+t-l would be an earlier repeat. Thus (1, a, ... , at-I) are all distinct, but at = 1. The integer t ~ 1 is called the order of a. This number will in general be different for different values of a; and given an element a, it may be difficult to calculate t. However, it turns out that we can say exactly how many elements of each order t ~ 1 are contained in F. Our first step in this direction is a special case of a famous theorem of Lagrange.