1.3. Proof of the Theorem

Try these first

1. 1.

   Define what is meant by the order of a finite field.

   Hint

   Think about how many elements there can be.

2. 2.

   What is the characteristic of a finite field?

   Hint

   Recall the prime numbers we discussed in class.

3. 3.

   What is the order of a finite field with characteristic 5 and r=3?

   • 15
   • 125
   • 5

   Hint

   Apply the formula we've discussed when calculating.

4. 4.

   True or False: The characteristic of a finite field can be a composite number.

   • True
   • False

   Hint

   Recall our definition of characteristic.

5. 5.

   Given a finite field of order 16, derive its characteristic and explain how this characteristic affects operations within the field.

   Hint

   Consider how powers of primes work in defining order.

6. 6.

   Using the irreducible polynomial x^2 + x + 1, construct a finite field and show all elements.

   Hint

   Use a systematic approach to list all possible combinations.