24. Finite Fields and Properties I

Try these first

1. 1.

   What is the characteristic of a finite field?

   Hint

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

2. 2.

   Can you give an example of a finite field?

   Hint

   Consider the operations defined among the integers.

3. 3.

   What is a finite field?

   • A field with infinitely many elements
   • A field with a finite number of elements
   • A field with real numbers

   Hint

   Think about the number of elements in the field.

4. 4.

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

   • True
   • False

   Hint

   Consider what we discussed about characteristics.

5. 5.

   Construct a finite field with 16 elements using polynomials, define its operations, and prove it satisfies field axioms.

   Hint

   Start with identifying irreducible polynomials over your coefficient field.

6. 6.

   If the characteristic of field F is 7, show that adding any element 7 times results in the additive identity.

   Hint

   Utilize the definition of characteristic directly.