Morphism from the Gaussian integers (2)

Let iC be the usual root of unity, with i2=1, and let Z[i]={a+bia,bZ} be the ring of Gaussian integers.

  1. Prove that there exists a (nonzero) ring homomorphism Z[i]Z5.
  2. Compute the kernel of your homomorphism explicitly, and state the conclusion given by the First Isomorphism Theorem. (Hint: The kernel requires two generators.)