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- Construct 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+.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]Prove that any field that contains an intergral domain D must contain a subfield isomorphic to the quotient field Q of D.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]
- 18. Let be the smallest subring of the field of rational numbers that contains . Find a description for a typical element of .True or False Label each of the following statements as either true or false. 4. Both , the set of even integers, and, the set of odd integers, are subrings of the set of all integers.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]
- 14. a. If is an ordered integral domain, prove that each element in the quotient field of can be written in the form with in . b. If with in , prove that if and only if in .Examples 5 and 6 of Section 5.1 showed that P(U) is a commutative ring with unity. In Exercises 4 and 5, let U={a,b}. Is P(U) a field? If not, find all nonzero elements that do not have multiplicative inverses. [Type here][Type here]8. Prove that the characteristic of a field is either 0 or a prime.