1.2. Properties of the Order of a Finite Field

Try these first

1. 1.

   What is the order of the finite field ℤ5?

   Hint

   Recall that the order is the number of elements.

2. 2.

   What is meant by the characteristic of a finite field?

   Hint

   Think about addition within the field.

3. 3.

   What is the order of a finite field with characteristic 7?

   • 7
   • 14
   • 7^2

   Hint

   Think about prime powers.

4. 4.

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

   • True
   • False

   Hint

   Consider what defines a field.

5. 5.

   Construct a finite field with characteristic 3 and of order 3^2.

   Hint

   Make sure your polynomial cannot be factored over ℤ.

6. 6.

   Show that the mapping from ℤr(p) to F is a bijection.

   Hint

   Use definitions of injectivity and surjectivity to support your argument.