1.1. Order of a Finite Field

Try these first

1. 1.

   Define the order of a finite field.

   Hint

   Think about how many elements exist in a given set.

2. 2.

   What is a prime characteristic?

   Hint

   Recall prime numbers and their properties.

3. 3.

   What does the order of a finite field represent?

   • The number of operations in the field
   • The total number of elements in the field
   • The degree of the field

   Hint

   Focus on what 'order' typically indicates in mathematics.

4. 4.

   Is it true that all finite fields must have prime order?

   • True
   • False

   Hint

   Recall the formula |F| = p^r.

5. 5.

   Given a prime p = 7 and r = 3, illustrate how to construct a finite field and explain the significance of irreducible polynomials in your construction.

   Hint

   Use the Euclidean method to check if your polynomial is irreducible.

6. 6.

   Prove that the mapping g from Z_p^r to the finite field is a bijection.

   Hint

   Construct the proof using closure and mapping properties of the elements.