1.4. Span of the Field

Try these first

1. 1.

   What is the order of a finite field with characteristic 5?

   Hint

   Remember the pattern of powers.

2. 2.

   Define what a minimal spanning set is.

   Hint

   Think about essential elements needed.

3. 3.

   What is the characteristic of a finite field?

   • A prime number
   • Any integer
   • A real number

   Hint

   Think about what defines the field's operations.

4. 4.

   True or False: The number of elements in a finite field can be any integer.

   • True
   • False

   Hint

   Recall the definition of order.

5. 5.

   Given p=5 and r=2, construct the finite field and demonstrate why the minimal spanning set must include at least two elements.

   Hint

   Focus on the relevance of irreducibility.

6. 6.

   Propose a general method for finding a minimal spanning set for any finite field of order p^r.

   Hint

   Consider linear combinations and independence.