Determine whether the given polynomial quotient ring R is a field or not. If R is a field, provide a proof. If not, provide a counterexample. (a) R = Z3[x] / (x3 + 2x2 + x + 1) (b) R = Z5[x] / (2x3 − 4x2 + 2x + 1) (c) R = Z2[x] / (x4 + x2 + 1)

Determine whether the given polynomial quotient ring R is a field or not. If R is a field, provide a proof. If not, provide a counterexample. (a) R = Z3[x] / (x3 + 2x2 + x + 1) (b) R = Z5[x] / (2x3 − 4x2 + 2x + 1) (c) R = Z2[x] / (x4 + x2 + 1)

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter8: Polynomials

Section8.6: Algebraic Extensions Of A Field

Problem 13E

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Question

Determine whether the given polynomial quotient ring R is a field or not. If R is a field,
provide a proof. If not, provide a counterexample.
(a) R = Z₃[x] / (x³ + 2x² + x + 1)

(b) R = Z₅[x] / (2x³ − 4x² + 2x + 1)

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