1.10. Field Definition and Operations

Try these first

1. 1.

   Define a finite field.

   Hint

   Think about what operations are allowed in a field.

2. 2.

   What is meant by the order of a finite field?

   Hint

   Recall the formula for expressing the order.

3. 3.

   The order of a finite field can be expressed as which of the following?

   • p^r
   • pr^2
   • 2p

   Hint

   Remember the definition of finite fields.

4. 4.

   True or False: The characteristic of a finite field can be any integer.

   • True
   • False

   Hint

   Think about the requirements for closure in addition.

5. 5.

   Demonstrate the construction of a finite field of order 16. Detail each step in the process.

   Hint

   Look for the irreducible polynomial x^4 + x + 1 as a starting point.

6. 6.

   Discuss and prove the relationship between a field's characteristic and its order using specific examples.

   Hint

   Use fields like F_3 and F_4 to demonstrate.