Let f: F→F be a nonzero homomorphism of a field F into itself. Show that f need not be onto (not surjective) Using a counterexample
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Let f: F→F be a nonzero homomorphism of a field F into itself. Show that f need not be onto (not surjective) Using a counterexample
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- Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero in[Type here] True or False Label each of the following statements as either true or false. 3. Every integral domain is a field. [Type here]Let where is a field and let . Prove that if is irreducible over , then is irreducible over .
- Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.Label each of the following as either true or false. If a set S is not an integral domain, then S is not a field. [Type here][Type here]Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].
- 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.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 .)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?
- 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.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]14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.