1.5. Minimal Spanning Set

Try these first

1. 1.

   Define the characteristic of a finite field.

   Hint

   Think about addition and how many times you reach zero.

2. 2.

   What does the order of a finite field represent?

   Hint

   Recall the relationship between characteristic and the number of elements.

3. 3.

   What is the characteristic of a finite field?

   • A composite number
   • A positive integer
   • A prime number

   Hint

   Consider the foundational concept of prime numbers.

4. 4.

   True or False: The order of a finite field is always a prime number.

   • True
   • False

   Hint

   Think about how orders are structurally defined with powers.

5. 5.

   Given a finite field F with characteristic 5 and order 25, demonstrate a minimal spanning set and explain how you know it's minimal.

   Hint

   Analyze the number of combinations needed to cover the entire field.

6. 6.

   Show that if F has characteristic p and has multiple minimal spanning sets, their cardinality must still equal each other.

   Hint

   Consider what it means for two sets to span the same field.