1A) Let F be a field and let f(x), g(x) and h(x) be polynomials in
F(x(. Prove that gcd(f(x),g(x)) = 1F and f(x) divides g(x)h(x), then
f(x) divides h(x).
1B) Let F be a field and let f(x), g(x), h(x), and d(x) be polynomials
in F(x(. Prove that if gcd(f(x),g(x)) = d(x) and both f(x) and g(x)
divide h(x) , then f(x)g(x) divides h(x)d(x).
Mathematics Homework Solutions
#75047
Fields, Polynomials, GCD and Divisibility
1A) Let F be a field and let f(x), g(x) and h(x) be polynomials in Fx. Prove that gcd(f(x),g(x)) = 1F and f(x) divides g(x)h(x), then f(x) divides h(x).
1B) Let F be a field and let f(x), g(x), h(x), and d(x) be polynomials in Fx. Prove that if gcd(f(x),g(x)) = d(x) and both f(x) and g(x) divide h(x) , then f(x)g(x) divides h(x)d(x).
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