20.2. Addition and Multiplication of Polynomials Over Fields

Try these first

1. 1.

   Define irreducible polynomial in simple terms.

   Hint

   Think about what would happen if we could break it down.

2. 2.

   In GF(5), what would be the result of adding (3x + 4) and (2x + 2)?

   Hint

   Be careful with adding coefficients!

3. 3.

   What is the result of adding 2x^2 + 2 and x^2 + 1 in GF(3)?

   • 1
   • 2x^2 + 2
   • 0

   Hint

   Remember to add coefficients considering modulo.

4. 4.

   True or False: The product of any two non-zero elements in a field cannot be zero.

   • True
   • False

   Hint

   Recall the definition of field behavior.

5. 5.

   If f(x) = x^4 + 1 and g(x) = x^2 + 2, determine their GCD over GF(5).

   Hint

   Start by dividing the higher degree polynomial.

6. 6.

   Prove that (x - 3) is a factor of f(x) = x^3 - 9 in GF(10).

   Hint

   Check for roots directly in the polynomial.