1.9. Existence of Irreducible Polynomials

Try these first

1. 1.

   Define what a finite field is.

   Hint

   Think about how many elements are included and what letters represent those elements.

2. 2.

   What defines an irreducible polynomial?

   Hint

   Consider its ability to be broken down into simpler polynomials.

3. 3.

   What is the characteristic of a finite field?

   • Any integer
   • An irrational number
   • A prime number

   Hint

   Consider what properties of numbers define a field's characteristic.

4. 4.

   True or False: Every polynomial can be factored over finite fields.

   • True
   • False

   Hint

   Think about polynomials that can and cannot be broken down further.

5. 5.

   Prove that the polynomial x^2 + 1 is irreducible over GF(5).

   Hint

   Evaluate the polynomial at each element of the field.

6. 6.

   Construct a field GF(2^3) using an irreducible polynomial.

   Hint

   Identify coefficients for the polynomial from GF(2) and how they contribute to the field.