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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]Prove Corollary 8.18: A polynomial of positive degree over the field has at most distinct zeros in
- 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]Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in 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.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]8. Prove that the characteristic of a field is either 0 or a prime.
