Let F be a field and a be a non-zero element in F. If f(x) is reducible over F, then f(x+a)EF[x] is None of the choices O Reducible O Unit O Irreducible
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Q: 10. Let F(a) be the field described in Exercise 8. Show that a² and a² + a are zeros of x³ + x + 1.
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- Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].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.Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero in
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- Suppose S is a subset of an field F that contains at least two elements and satisfies both of the following conditions: xS and yS imply xyS, and xS and y0S imply xy1S. Prove that S is a field. This S is called a subfield of F. [Type here][Type here]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 .If 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 .
- Label each of the following statements as either true or false. Every f(x) in F(x), where F is a field, can be factored.Each of the polynomials in Exercises is irreducible over the given field . Find all zeros of in the field obtained by adjoining a zero of to . (In Exercises and , has three zeros in .)