Exercise 4.2. (Bezout's identity for polynomials) Let F be a field. Let f, g e F[x] with greatest common divisor d. Prove that there exist polynomials u and v such that fu+gv = d.
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![Exercise 4.2. (Bezout's identity for polynomials) Let F be a field. Let f, g E F[x] with
greatest common divisor d. Prove that there exist polynomials u and v such that fu+gv = d.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf768e71-e1cd-4ccd-807c-459a250f7c7a%2Fe1a8e896-9d88-45b5-99fc-b854ccb387db%2Fjreg27e_processed.png&w=3840&q=75)
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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]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 overSuppose 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 ofLet be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero inLet 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.
- Since this section presents a method for constructing a field of quotients for an arbitrary integral domain D, we might ask what happens if D is already a field. As an example, consider the situation when D=5. a. With D=5, write out all the elements of S, sort these elements according to the relation , and then list all the distinct elements of Q. b. Exhibit an isomorphism from D to Q.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 .Prove that if F is an ordered field with F+ as its set of positive elements, then F+nen+, where e denotes the multiplicative identity in F. (Hint: See Theorem 5.34 and its proof.) Theorem 5.34: Well-Ordered D+ If D is an ordered integral domain in which the set D+ of positive elements is well-ordered, then e is the least element of D+ and D+=nen+.