Show that any root of 1 + x + x 6 ∈ F 2 [ x ] is a primitive element of F 64 . Consider the field with XXXXXXXXXXelements constructed using the irreducible polynomial f ( x ) = 1 + x 3 + x 4 over F 2...

1. 
   

   
   Show that any root of 1 +
   
   x
   
   
   
   +
   
   x
   
   
   
   
   ⁶
   
   
   
   ∈
   
   F
   
   
   ₂
   
   [
   
   x
   
   
   
   ] is a primitive element of
   
   F
   
   
   ₆₄
   
   .
   

   
   


2. 
   

   
   Consider the field with 16 elements constructed using the irreducible polynomial
   
   f
   
   
   
   (
   
   x
   
   
   
   ) = 1 +
   
   x
   
   
   
   
   ³
   
   
   
   +
   
   x
   
   
   
   
   ⁴
   
   
   
   over
   
   F
   
   
   ₂
   
   .
   

   
   
   
   
   1. 
      

      
      Let
      
      α
      
      be a root of
      
      f
      
      (
      
      x
      
      ). Show that
      
      α
      
      is a primitive element of
      
      F
      
      
      ₁₆
      
      .
      

      
      
   
   
   
   

Represent each element both as a polynomial and as a power of

α

.

1. 
   

   
   Construct a Zech’s log table for the field.
   

   
   

Q6

1. 
   

   
   Find a primitive element and construct a Zech’s log table for each of the following finite fields:
   

   
   

(a)

F


₃


²



, (b)

F


₂


⁵



, (c)

F


₅


²



.