et FCK be a field extension and let R be the alg K. Then R is a subfield of K and FCR.
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- Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros inProve that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.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+.
- 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.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 over
- 18. Let be the smallest subring of the field of rational numbers that contains . Find a description for a typical element of .8. Prove that the characteristic of a field is either 0 or a prime.Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)
- Prove that a polynomial f(x) of positive degree n over the field F has at most n (not necessarily distinct) zeros in F.Let S be a nonempty subset of an order field F. Write definitions for lower bound of S and greatest lower bound of S. Prove that if F is a complete ordered field and the nonempty subset S has a lower bound in F, then S has a greatest lower bound in F.Prove Theorem If and are relatively prime polynomials over the field and if in , then in .