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- 14. Prove or disprove that is a field if is a field.Consider the set ={[0],[2],[4],[6],[8]}10, with addition and multiplication as defined in 10. a. Is R an integral domain? If not, give a reason. b. Is R a field? If not, give a reason. [Type here][Type here]8. Prove that the characteristic of a field is either 0 or a prime.
- Consider the set S={[0],[2],[4],[6],[8],[10],[12],[14],[16]}18, with addition and multiplication as defined in 18. a. Is S an integral domain? If not, give a reason. b. Is S a field? If not, give a reason. [Type here][Type here]Prove that if R and S are fields, then the direct sum RS is not a field. [Type here][Type here]Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.
- Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros inConstruct a field having the following number of elements.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+.
