24.3.2. Theorem on Characteristic of Finite Fields

Try these first

1. 1.

   What is the characteristic of a finite field?

   Hint

   Think about how many times you have to add 1 to get back to 0.

2. 2.

   Give an example of a field and its characteristic.

   Hint

   Check what happens when you add 1 repeatedly.

3. 3.

   What is the characteristic of a field?

   • The smallest prime number
   • The smallest positive integer such that m times 1 = 0
   • The number of elements in the field

   Hint

   Think about how you derive 0 from repeatedly adding 1.

4. 4.

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

   • True
   • False

   Hint

   Consider the implications of composite numbers.

5. 5.

   Create a finite field with a characteristic of a composite number and demonstrate why it leads to a contradiction.

   Hint

   Identify the meaning behind composite numbers and their factors.

6. 6.

   Investigate the properties of a finite field of characteristic 7 and discuss its closure under operations.

   Hint

   Each operation shouldn't produce an element outside of 0-6.