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)?
1
Expert's answer
2012-06-15T12:00:59-0400
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]
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