ついでに解答は
Imf = F(Z[x]) = Z[√3] = {a+b√3|a,b∈Z}
Kerf = {f(x)∈Z[x]|f(√3)=0} = (x^2-3) = {(x^2-3)g(x)|g(x)∈Z[x]}
でおｋ？