Exercise 11.3. Let F be a finite field, and F(F) be the ring of functions from F to F. Show that the ring homomorphism : F[x] → F(F) defined by is surjective. (p(x))(a) := p(a), VaЄ F, p(x) = F[x],
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![Exercise 11.3. Let F be a finite field, and F(F) be the ring of functions from F to F. Show that
the ring homomorphism : F[x] → F(F) defined by
is surjective.
(p(x))(a) := p(a), VaЄ F, p(x) = F[x],](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff2e948f6-fd6f-485f-942e-c931230f8579%2Ff2e8476b-5843-4aa7-b59c-59f65422ceb6%2Fo22cjor_processed.jpeg&w=3840&q=75)
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- 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 .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 ab in a field F. Show that x+a and x+b are relatively prime in F[x].11. Show that defined by is not a homomorphism.Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove that is a ring homomorphism. Is (e)=e where e is the unity in 3 and e is the unity in 12?
- 14. Prove or disprove that is a field if is a field.Let R be a commutative ring with unity. Prove that deg(f(x)g(x))degf(x)+degg(x) for all nonzero f(x), g(x) in R[ x ], even if R in not an integral domain.Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]
- 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]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 .)Consider the set of matrices 0 S = {(a+b):abER} a, bЄR