24.1. Construction of Finite Field with 9 Elements

Try these first

1. 1.

   Define finite field and give an example.

   Hint

   Think about the number of elements in the field.

2. 2.

   What is an irreducible polynomial?

   Hint

   Consider how polynomials behave similar to numbers.

3. 3.

   What does the characteristic of a finite field represent?

   • The number of elements
   • The size of the field
   • The smallest integer m such that m times 1 = 0

   Hint

   Think about what happens if you keep adding 1.

4. 4.

   Is the characteristic of a finite field always a prime number?

   • True
   • False

   Hint

   Consider the implications of composite numbers.

5. 5.

   Construct the polynomial multiplication table for the elements of F and determine the product of (x + 1)(x + 2).

   Hint

   Remember to reduce after multiplication and apply the modulo.

6. 6.

   Using F, show that multiplying each polynomial by the element 2 generates a cyclic group. List the non-zero elements produced.

   Hint

   Keep track of each step, reducing properly along the way.