In the ring Z[a], let I = (x³ – 8).(a) Let f(x) = 4.x³ + 6x4 – 2x³ +x² – 8x +3 € Z[r]. Find a polynomial p(x) E Z[r] such thatdeg p(x) < 2 and f(x) = p(x) (mod I).(b) Prove that the quotient ring Z[r]/I is not an integral domain.

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.3: Factorization In F [x]

Problem 8E: Let be an irreducible polynomial over a field . Prove that is irreducible over for all nonzero ...

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In the ring Z[r], let I = (x³ – 8). (a) Let f(r) = 4rð + 6x4 – 2a³ + x² – 8x +3 € Z[r). Find a polynomial p(x) E Z[r] such that deg p(x) < 2 and f(x) = p(x) (mod I). (b) Prove that the quotient ring Z[r]/I is not an integral domain.