Ring Theory - The ring of Gaussian integers.

Let R=ℤ[i]={a+bi with a,b in ℤ} be the ring of Gaussian integers.

(a) Show that I={α(4-1) with α in ℝ} is an ideal of R.

(b) Show that ϕ: R ℤ/17ℤ, given by ϕ(a+bi)=a+4b, is a surjective ring homomorphism.

(c) Now show that R/I is isomorphic with ℤ/17ℤ (as rings).