Answer to Question #5108 in Abstract Algebra for Annemarie

Question

Let Z be the ring of integers. Consider the function
f : Z[x] → Z
defined by f(g(x)) = g(0). for example, f(x2 + 1) = 1.
(a) Show that f is a ring homomorpism.
(b) What is the ker(f)?

Answer 1

1) we must show that f is save multiplication and addition
let h(x)=
am*x^m+...+a0,
g(x)=bn*x^n+...+b0
f(h(x)+g(x))=[h(x)+g(x)](0)=h(0)+g(0)=f(h(x))+f(g(x))
f(h(x)*g(x))=f(am*bn*x^(n+m)+...+a0*b0)=am*bn*0^(n+m)+...+a0*b0=a(0)*b(0)=f(h(x))*f(g(x))

so
f is ring homomorphism
2) Ker(f) all that polynoms such that g(0)=0 so it
have g(x)=x*h(x), where h - polynom from Z[x]