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Q: 2. Prove that in any ordered field F, a? + 1 > 0 for all a e F. Conclude that any field with a…
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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 F be an infinite field and let f(x), g(x) E F[x]. If f(a) = g(a) for
infinitely many elements a of F, show that f(x) = g(x).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F218b2c9d-de4c-4305-9fe2-eec55d5fa393%2Fb3dc1411-f10f-4790-b8ee-d90c79c06cb8%2Fipx2nc.jpeg&w=3840&q=75)
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- 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.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 a field. Prove that if is a zero of then is a zero of
- 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]Let where is a field and let . Prove that if is irreducible over , then is irreducible over .If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type here][Type here]
- 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 .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 .)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 .