# Homework 2

Due: Monday, September 27, 1999

• 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 constructa 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 constructa 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.  A degree 4 irreducible polynomial p(x) in Peterson's Table of Irreducible Polynomials over
```                       DEGREE 4  ...  3  37D  ...
```
• What is p(x)? p(x)= ...
• Since p(x) is irreducible and of degree 4, 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).
• Let ksi = x mod p(x). Why is { ksik} not a complete list of all the non-zero elements of GF(24)?