1.11. Existence of Multiplicative Inverse

Try these first

1. 1.

   What is the characteristic of a finite field?

   Hint

   Think about how often you need to add 1 before you get 0.

2. 2.

   Define a multiplicative inverse.

   Hint

   Can you think of any examples of such elements?

3. 3.

   What is the characteristic of a finite field?

   • A prime number
   • Any integer
   • Only even integers

   Hint

   Remember that characteristics involve addition leading to zero.

4. 4.

   True or False: Every non-zero element in a finite field has a unique multiplicative inverse.

   • True
   • False

   Hint

   Think about the definitions we've discussed.

5. 5.

   Given a finite field with characteristic 5, find the order of the field if r = 3. Prove the validity of your answer.

   Hint

   Remember the form pr for field order.

6. 6.

   Construct a finite field from the polynomial x^2 + 1 over GF(3). Identify all elements and calculate the multiplicative inverse of 2.

   Hint

   Utilize the polynomial to generate elements and assess their relationships.