# Let F be a field and let a(X), b(X), c(X) and p(X) bâ€¦ Show

Let F be a field and let a(X), b(X), c(X) and p(X) bâ€¦ Show more guys, please help with modern algebra homework? Let F be a field and let a(X), b(X), c(X) and p(X) be polynomials in F[X]. Prove that if gcd(a(X), c(X)) = 1F and gcd(a(X), b(X)) = p(X), then gcd(a(X), b(X)c(X)) = p(X). Let F be a field and let e(X), f(X),g(X) and n(X) be polynomials in F[X] with n(X) of positive degree. Prove that if e(X) = gcd( f (X), n(X)) and e(X) divides g(X), then there is a polynomial j(X) âˆˆ F[X] such that f (X) j(X) â‰¡ g(X) (mod n(X)). Let F be a field and let a(X), b(X), c(X), d(X), f (X) and g(X) be a polynomials where d(X), f (X) and g(X) have positive degrees. Prove that if a(X) â‰¡ b(X) (mod f (X)), b(X) â‰¡ c(X) (mod g(X)) and gcd( f (X), g(X)) = d(X), then a(X) â‰¡ c(X) (mod d(X)). Let R be the set of matrices of the form a 0 where a, c and d are real numbers. Prove that the set cd I = AâˆˆR|A= 0 0 for some real numbers f and g is an ideal of R. â€¢ Show less

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