1.6. Mapping from ℤ^r to Field F

Try these first

1. 1.

   What is the order of a finite field?

   Hint

   Think about the prime number's role in determining the number of elements.

2. 2.

   Define a minimal spanning set.

   Hint

   Consider redundancy in spanning the field.

3. 3.

   What is the order of a finite field?

   • p^r
   • r^p
   • pq
   • None of the above

   Hint

   Focus on the relationship between the prime and the number of elements.

4. 4.

   True or False: Every polynomial can be used to construct a finite field.

   • True
   • False

   Hint

   Consider the properties needed for a polynomial to work for fields.

5. 5.

   Construct a finite field of order 16 and describe the process involved in selecting the irreducible polynomial.

   Hint

   Keep in mind the necessary properties of irreducible polynomials.

6. 6.

   Demonstrate that for any finite field with order p^r, the mapping function g is bijective.

   Hint

   Refer back to the properties of linear combinations and minimal spanning sets.