Exercise 13.6. Let : F → R be a ring homomorphism where F is a field and OR 1R. Show that the characteristic of F is equal to the characteristic of R.
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- Suppose that f(x),g(x), and h(x) are polynomials over the field F, each of which has positive degree, and that f(x)=g(x)h(x). Prove that the zeros of f(x) in F consist of the zeros of g(x) in F together with the zeros of h(x) in F.Let be a field. Prove that if is a zero of then is a zero ofLet R be a commutative ring with unity. Prove that deg(f(x)g(x))degf(x)+degg(x) for all nonzero f(x), g(x) in R[ x ], even if R in not an integral domain.
