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A: By the method of contradiction i have solved the problem
![Compute the gcd of the following polynomials in Z(x]
and
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- Find all monic irreducible polynomials of degree 2 over Z3.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 in
- 22. Let be a ring with finite number of elements. Show that the characteristic of divides .Find a principal ideal (z) of such that each of the following products as defined in Exercise 10 is equal to (z). a. (2)(3)(4)(5)(4)(8)(a)(b)Prove Theorem If and are relatively prime polynomials over the field and if in , then in .
- 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.Prove Theorem Suppose is an irreducible polynomial over the field such that divides a product in , then divides some .