Give a proof for Corollary 3.3 in Section 3.2. In other words, prove that in any finite field of characteristic p,

for any m ? 1.

Let GF(2 ⁶) be defined by the primitive polynomial f( x) = x ⁶ + x + 1 and let ? be a root of f( x). Compute: ? ⁹ + ? ²³, ? ⁹ ? ²³, and 1 + ? ⁷.

Let p be a prime number. Prove that

The cyclotomic coset containing s consists of

where n _s is the smallest positive integer such that (mod p ⁿ 1). Prove that n _s n.

Let ? be a primitive element in GF( p ⁿ). Prove that m _s( x) defined in the algorithm in Section 3.4 is a polynomial over GF( p). In other words, if

where C _s is the cyclotomic coset modulo p ⁿ 1 containing s as the coset leader, then the coefficients of m _s( x) belong to GF( p).

Let GF(2 ⁵) be defined by the primitive polynomial f( x) = x ⁵ + x ³ + 1 and let ? be a root of f( x).

1. Compute...