1. Finite Fields and Properties II

Try these first

1. 1.

   What does the order of a finite field indicate?

   Hint

   Think about the definition we explored in our discussions.

2. 2.

   Define an irreducible polynomial.

   Hint

   Consider the properties of polynomial expressions.

3. 3.

   What does the order of a finite field represent?

   • The total number of elements in the field
   • The maximum degree of polynomials
   • The characteristic of the field

   Hint

   Think back to the definition we studied.

4. 4.

   True or False: The characteristic of a finite field is always a composite number.

   • True
   • False

   Hint

   Recall our definition of characteristic.

5. 5.

   Prove that for any finite field F with characteristic p, the order is of the form p^r for some integer r > 0.

   Hint

   Investigate how the mapping and linear combinations play a role.

6. 6.

   Construct the finite field of order 16 using an irreducible polynomial and verify the properties hold.

   Hint

   Check the polynomial functions to ensure they adhere to defined field properties.