Prove or disprove Let K be an extension of a field F and a ∈ K be algebraic over F. Then F[a] = F (a), where F[a] = {f(a): f(x) ∈ F [x]}. , Please do not copy
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A: Hello, learner we can answer first question as per the honor policy. Please resubmit other question…
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Prove or disprove Let K be an extension of a field F and a ∈ K be algebraic over F. Then F[a] = F (a), where F[a] = {f(a): f(x) ∈ F [x]}. , Please do not copy
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- Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros inProve Theorem If and are relatively prime polynomials over the field and if in , then in .Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.
- Let S be a nonempty subset of an order field F. Write definitions for lower bound of S and greatest lower bound of S. Prove that if F is a complete ordered field and the nonempty subset S has a lower bound in F, then S has a greatest lower bound in F.Let F be a field and f(x)=a0+a1x+...+anxnF[x]. Prove that x1 is a factor of f(x) if and only if a0+a1+...+an=0. Prove that x+1 is a factor of f(x) if and only if a0+a1+...+(1)nan=0.True or False Label each of the following statements as either true or false. 4. Any polynomial of positive degree over the field has exactly distinct zeros in .
- Prove Theorem Suppose is an irreducible polynomial over the field such that divides a product in , then divides some .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 where is a field and let . Prove that if is irreducible over , then is irreducible over .