Let σm: ℤ → ℤm be the natural homomorphism given by σm (a) = (the remainder of a when divi...

Let σm: ℤ → ℤm be the natural homomorphism given by σm (a) = (the remainder of a when divided by m) for a ∊ ℤ.

a. Show that : ℤ [x] → ℤm [x] given by

(a0+a1x+...+anxn) = + σm (a1)x +... + σm (an)xn

is a homomorphism of ℤ [x] onto ℤm [x].

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

c. Use part to show that x3 + 17x + 36 is irreducible in ℚ [x]. [Hint: Try a prime value of m that simplifies the coefficients.]