24.2. Verification of Field Axioms

Try these first

1. 1.

   What is the characteristic of a finite field?

   Hint

   Think about how many times you can add 1 to reach 0.

2. 2.

   What is an irreducible polynomial?

   Hint

   Consider the definitions of factorization.

3. 3.

   What is the characteristic of a finite field?

   • Any integer
   • Only prime numbers
   • Negative numbers
   • Zero

   Hint

   Consider the definition of characteristic.

4. 4.

   True or False: An irreducible polynomial can be factored into simpler polynomials.

   • True
   • False

   Hint

   Think about what 'irreducible' implies.

5. 5.

   Prove that if you have a finite field with characteristic p, where p is prime, can you show that there are no non-trivial divisors of p?

   Hint

   Review the definition of prime numbers.

6. 6.

   Given an irreducible polynomial of degree 2, how would you construct a finite field using it?

   Hint

   Recall the construction process we discussed in class.