Let F be a field and let f(z) F[r]. Suppose deg(f(x)) = n 0. F[r]/(f(x))...

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Let F be a field and let f(z) F[r]. Suppose deg(f(x)) = n ≥ 0. F[r]/(f(x)) is a vector space over F. Prove that F[z]/(f(x)) has dimension n by showing that {1,2,²,,-1) is a basis. Remark: This problem shows that every element of F[x]/(f(x)) can be represented in a unique way by a polynomial of degree less than deg(f(x)). Let F be a field and let f(z) F[r]. Suppose deg(f(x)) = n ≥ 0. F[r]/(f(x)) is a vector space over F. Prove that F[z]/(f(x)) has dimension n by showing that {1,2,²,,-1) is a basis. Remark: This problem shows that every element of F[x]/(f(x)) can be represented in a unique way by a polynomial of degree less than deg(f(x)).