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4. (30) Let C be the ring of complex numbers,and letf:C→C be the map defined by...

4. (30) Let C be the ring of complex numbers,and letf:C→C be the map defined by

f(z) = z^3.

  1. (i) Prove that f is not a homomorphism of rings, by finding an explicit counterex-

    ample.

  2. (ii) Prove that f is not injective.

  3. (iii) Prove that the principal ideal I = 〈x^2 + x + 1〉 is not a prime ideal of C[x].

(iv) Determine whether or not the ring C[x]/I is a field.

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