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Exercises using none topaint none with web,windows application Java 10.01 Factor x 3 x into linear fac tors in F3[X]. 10.02 Factor x into linear factors in Fs[x].

(ans.). S + x + 1 into irreducibles in F2[X], by trial division.

(am.) 10.03 Factor x 10.

04 Factor X S + x4 + 1 into irreducibles in F 2 [xJ by trial division. 10.05 Factor x 6 + x 3 + X + 1 into irreducibles in F2[X] by trial division.

(am) 10.06 Let k[x] be the polynomial ring in one variable x over the field k. What is the group of units k[x] x (meaning the collection of polynomials that have multiplicative inverses which are also polynomials) 10.

07 Find the greatest common divisor of Xli + x4 + x 3 + x 2 + X + 1 and , X4 + x2 + 1 in the ring Q [x] of polynomials over Q. (am.).

XS -. 10.08 Find the gr eatest common divisor of x 6 + x 3 + 1 and x 2 + x + 1 in the ring k[x] of polynomials over the finite field k = Z/3 with 3 elemeQts. 10.

09 Find the greatest common divisor of the two polynomials x 6 + x4 + x 2 + 1 and x8 + x 6 + x4 + x 2 + 1 in the ring k[x] of polynomials over th~ finite field k = Z/2 with 2 elements. 10.10 Find the greatest common divisor ofthe two polynomials x S +x+l and X S + x4 + 1 in the polynomial ring F 2 [x].

(ana.). Exercises 10.11 Find the gr eatest common divisor of the two polynomials x 5 +X4 +X3 +1 and x 5 + x 2 + X + 1 in F 2 [x}. 10.

12 Find the greatest common divisor of x 7 + x 6 + x 5 + x4 + 1 and x 6 +x5 + x4 +x3 + x 2 +x + 1 in F 2 [x}. (ana.) 10.

13 Find the greatest common divisor of x 5 + x 3 + x 2 + 1 and x 6 + x 5 + X + 1 in F2[X}.. Fi nite Fields 11.1 11.2 11.

3 11.4 11.5.

Making fields Exa none none mples of field extensions Addition mod P Multiplication mod P Multiplicative inverses mod P. Again, while we a re certainly accustomed to (and entitled to) think of the fields rationals, reals, and complex numbers as "natural" batches of numbers, it is important to realize that there are many other important and useful fields. Perhaps unexpectedly, there are many finite fields: For example, for a prime number p, the quotieBt Zip is a field (with p elements). On the other hand, for example, there is no finite field with 6 or with 10 elements.

(Why ) While it turns out that there are finite fields with, for example, 9 elements, 128 elements, or any prime power number of elements, it requires more preparation to "find" them. The simplest finite fields are the rings Zip with p prime. For many different reasons, we want more finite fields than just these.

One immeqiate reason is that for machine implementation (and for other computational simplifications) it is optimal to use fields of characteristic 2, that is, in which 1 + 1 = 2 = O. Among the fields Zip only Z/2 satisfies this condition. At the same time, for various reasons we might want the field to be larye.

H we restrict our attention to the fields Zip we can"t meet both these conditions simultaneously.. 11.1 Making fields Construction of f inite fields, and computations in finite fields, are based upon polynomial computations. 192. Making fields For brevity, writ e F q for the finite field with q elements (if it exists!) For a prime p at least we have one such finite field, namely Zip = F p" Again, another notation often seen is GF(q) = Fq Here "GF" stands for Galois field. Remark: There is the issue of uniqueness of a finite field with a given number of elements. It is true that there is essentially at most one such, but this is not easy to prove.

Also, in practice the various possible computational models of the same underlying abstract object have a great impact, so we will often be more concerned with the many different models themselves. Remark: The present discussion continues to be entirely analogous to our discussion of Z/m. For a polynomial P (not necessarily irreducible), and for two other polynomials f, g, all with coefficients in F P" write.

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