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Elements Of Modern Algebra

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.1: Polynomials Over A Ring

Problem 14E: 14. Prove or disprove that is a field if is a field.

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14. Prove or disprove that is a field if is a field.
If is a finite field with elements, and is a polynomial of positive degree over , find a formula for the number of elements in the ring .
Let ab in a field F. Show that x+a and x+b are relatively prime in F[x].
Construct a field having the following number of elements.
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]
Label each of the following statements as either true or false. Every f(x) in F(x), where F is a field, can be factored.
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]
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]

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