1.7. Proof of Bijection

Try these first

1. 1.

   What is the order of a finite field?

   Hint

   Think about how we describe the size of a mathematical structure.

2. 2.

   What does the characteristic of a finite field represent?

   Hint

   Consider how many times you can add one before returning to zero.

3. 3.

   What is the characteristic of a finite field?

   • A prime number
   • A composite number
   • A non-integer

   Hint

   Recall the definition of field characteristics.

4. 4.

   True or False: The elements of a minimal spanning set are always unique.

   • True
   • False

   Hint

   Think about redundancy in spans.

5. 5.

   Construct a finite field of order 49 and explain the choice of irreducible polynomial.

   Hint

   Consider how the degree of the irreducible polynomial corresponds to the structure of the finite field.

6. 6.

   If given a finite field of order 9, demonstrate how to show that it is constructed using an irreducible polynomial.

   Hint

   What properties must the polynomial possess to be irreducible in this context?