Please answer only no. 8Show complete solution to the given problem.Thank you! 7.)a. Prove that= {a, + apx + *. + ar" |a, = 2k for k E Z),%3D%3Dthe set of all polynomials in Zx]with even constant term, is an ideal of Z[x).b. Show that /[x] = {x•f(x) + 2 g(x)| f(x), g(x) E Z]}.8. a. Prove thatI[x] = {do + a* + ** + a"|a, 2k, for k,E Z),2k, for k, E Z),%3D%3Dthe set of all polynomials with even coefficients, is an ideal of Z[x.b. Prove or disprove that /Ex is a principal ideal.9. Ja. Let F be a field. Prove thatI[x] = {ao + ax +*** + a,r"|a, E Fand a, + a, + + a, = 0},the set of all polynomials in F[x] such that the sum of the coefficients is zero, is anideal of F[x]....%3D%3Db. Prove or disprove that /[x] is a principal ideal.

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Answered: 8. a. Prove that I[x] = {do + ax + +…

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 8E

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8. a. Prove that I[x] = {do + ax + + a,x"|a, = 2k, for k, E Z), %3D %3! ... the set of all polynomials with even coefficients, is an ideal of Z[x. b. Prove or disprove that /[x] is a principal ideal.