1.12. Final Construction of Fields

Try these first

1. 1.

   Define the characteristic of a finite field.

   Hint

   Think about how addition behaves in modular arithmetic.

2. 2.

   What does the order of a finite field refer to?

   Hint

   It involves both the prime number and the value of r.

3. 3.

   What is the characteristic of a finite field?

   • Any integer
   • A prime integer
   • A composite integer

   Hint

   Think about which types of integers can sum to zero in a specific context.

4. 4.

   The order of a finite field can be expressed as:

   • True
   • False

   Hint

   Consider how we define field sizes.

5. 5.

   Construct a finite field of order 9 and detail your steps.

   Hint

   Identify possible polynomials and ensure they meet irreducibility traits.

6. 6.

   Prove why fields constructed with non-irreducible polynomials cannot form valid finite fields.

   Hint

   Consider how irreducibility affects factor structures and field axioms.