# Homework 3

• MacWilliams & Sloane, Chap. 4
• Peterson & Weldon, Chap. 6

• Problem 1.  The polynomial

p(x) = x6 + x5 + 1

is primitive (hence, irreducible) over GF(2) . Use p(x) to construct a log/antilog table for GF(26) .

• Problem 2.  The polynomial

p(x) = x2 + x + 2

is primitive (hence, irreducible) over GF(3) . Use p(x) to construct a  log/antilog table for GF(32) .

• Problem 3.  Let  R   be a commutative ring with non-zero elements  a  and  b  such that

ab = 0

Prove that  is not a field.

• Problem 4.  Consider the following degree 4 irreducible polynomial p(x) given in Peterson's Table of Irreducible Polynomials over GF(2)
` `
`                     DEGREE 4  ...  3  37D  ...`
• What is p(x)?    p(x)=   .…….

o       Since p(x) is irreducible and of degree 3, it follows that

GF(24) = GF(2)[x] mod p(x)

List all the elements of GF(24) in the above representation, i.e., in terms of

ksi = x mod p(x).

o       Let ksi = x mod p(x). Why is { ksik} not a complete list of all the non-zero elements of GF(24)?