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Fill in the blank:
1. The remainder when f(x) in R[x] below is divided by g(x) in R[x] is equal to ____.
f(x) = x^2022 - 1
g(x) = x + 1
2. The quotient ring F[x]/(f) is a field iff f is _____.
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- True or False Label each of the following statements as either true or false. 8. Any polynomial of positive degree that is reducible over a field has at least one zero in .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 Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in
- 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.Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.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 be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero inProve Theorem If and are relatively prime polynomials over the field and if in , then in .Prove Theorem Suppose is an irreducible polynomial over the field such that divides a product in , then divides some .
- True or False Label each of the following statements as either true or false. Every polynomial equation of degree over a field can be solved over an extension field of .Corollary requires that be a field. Show that each of the following polynomials of positive degree has more than zeros over where is not a field. over overIf is a finite field with elements, and is a polynomial of positive degree over , find a formula for the number of elements in the ring .