Let m : Z Z m be the natural homomorphism given by m (a)

Let σ_m : Z → Z_m be the natural homomorphism given by σ_m(a) = (the remainder of a when divided by m) for a ∈ Z. 

a. Show that σ̅_m : Z[x] → Z_m[x] given by σ̅_m(a₀ + a₁x + · · · + a_nxⁿ) = σ_m(a₀) + σ_m(a₁)x + · · · + σ_m(a_n)xⁿ is a homomorphism of Z[x] onto Z_m[x].

b. Show that if f(x) ∈ Z[x] and σ̅_mf(x)) both have degree n and σ̅_m f(x)) does not factor in Z_m[x] into two polynomials of degree less than n, then f(x) is irreducible in Q[x]. 

c. Use part (b) to show that x³ + I7x + 36 is irreducible in Q[x].