1. In the ring Z[], let I = (r³ – 8).(a) Let f(x) = 4r³ + 6x* – 2x³ + x² – 8x +3 € Z[r]. Find a polynomial p(x) E Z[r] such thatdeg p(r) < 2 and f(x) = p(x) (mod I).

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Answered: 1. In the ring Z[r], let I = (r* – 8).…

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 20E: Prove Theorem If and are relatively prime polynomials over the field and if in , then in .

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1. In the ring Z[r], let I = (r* – 8). (a) Let f(r) = 4r³ + 6x* – 2r³ +x² – 8r+3 € Z[r]. Find a polynomial p(r) E Z[z] such that deg p(r) < 2 and f(r) = p(x) (mod I).