Please explain carefully.

4. (a) Find all ring homomorphisms Q[x] C. (Hint: Is such a homomorphism deter- mined by the image of x?) (b) Let R and S be rings and I an ideal of R. Let a : R + R/I be the quotient map. Let 0 : R + S be a ring homomorphism. Show that I c ker(0) if and only if there is a ring homomorphism 0 : R/I S such that 0 = 0 on. Moreover, show that the map is unique when it exists. Conclude that we have a bijection {0 Hom(R, S):I C ker(0)} + Hom(R/I, S) given by 0 H 0. (See the notes at the beginning for the notation Hom(R, S).) (c) Find all ring homomorphisms Q[x]/(x 2) + C and Q[x]/(x 2) + R and the kernel and image of each. Are the images fields? (Hint: Is x3 2 irreducible in Q[x]?) (d) Find all ring homomorphisms Q[x]/(x3 8) + C and the kernel and image of each. 4. (a) Find all ring homomorphisms Q[x] C. (Hint: Is such a homomorphism deter- mined by the image of x?) (b) Let R and S be rings and I an ideal of R. Let a : R + R/I be the quotient map. Let 0 : R + S be a ring homomorphism. Show that I c ker(0) if and only if there is a ring homomorphism 0 : R/I S such that 0 = 0 on. Moreover, show that the map is unique when it exists. Conclude that we have a bijection {0 Hom(R, S):I C ker(0)} + Hom(R/I, S) given by 0 H 0. (See the notes at the beginning for the notation Hom(R, S).) (c) Find all ring homomorphisms Q[x]/(x 2) + C and Q[x]/(x 2) + R and the kernel and image of each. Are the images fields? (Hint: Is x3 2 irreducible in Q[x]?) (d) Find all ring homomorphisms Q[x]/(x3 8) + C and the kernel and image of each