Question

(abstract algebra) Let F be a field. Suppose f(x), g(x), h(x) ∈ F[x]. Show that the following properties hold:

(a) If g(x)|f(x) and h(x)|g(x), then h(x)|f(x).

(b) If g(x)|f(x), then g(x)h(x)|f(x)h(x).

(c) If h(x)|f(x) and h(x)|g(x), then h(x)|f(x) ± g(x).

(d) If g(x)|f(x) and f(x)|g(x), then f(x) = kg(x) for some k ∈ F \ {0}

Answer 1

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