4. Prove that every finite field is a homomorphic image of Z[x].
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- Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.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.Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.
- 8. Prove that the characteristic of a field is either 0 or a prime.Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]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.
- Use Theorem to show that each of the following polynomials is irreducible over the field of rational numbers. Theorem Irreducibility of in Suppose is a polynomial of positive degree with integral coefficients and is a prime integer that does not divide. Let Where for If is irreducible in then is irreducible in .Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero inIf 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 .