Let f(x) = x® +x³ + 1. (a) Prove that f(x) is reducible over Z3. Write f(x) as a product of irreducible factors. (b) Prove that f (x) is irreducible over Z2. (c) Let I = (f(x)) be the principal ideal...

Answer C and D only.

Extracted text: Let f(x) = x® +x³ + 1. (a) Prove that f(x) is reducible over Z3. Write f(x) as a product of irreducible factors. (b) Prove that f (x) is irreducible over Z2. (c) Let I = (f(x)) be the principal ideal generated by f(x) in Z2[x]. Calculate the multi- plicative inverse of (x³ + 1) + I in Z2[x]/I. %3D (d) Use part (b) to prove that g(x) (1/6)r6 + (2/3)³+ (4/3)aª + (7/6)x³ + (5/3)x² + (2/3)x + (5/6) is irreducible over Q.