Answered: 1.29. Let f = x² + x + 1. (a) Is the…

1.29. Let f = x² + x + 1. (a) Is the ring F7[x]/(f) an integral domain? (b) Show that Z[x]/(7) = F7[x]. (c) Is (f, 7) a maximal ideal of Z[x]? Is it a prime ideal?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.2: Integral Domains And Fields

Problem 17E: If e is the unity in an integral domain D, prove that (e)a=a for all aD. [Type here][Type here]

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Question

18.1.29. Let f = x² + x + 1.
(a) Is the ring F7[x]/(f) an integral domain?
(b) Show that Z[x]/(7) = F7[x].
(c) Is (f, 7) a maximal ideal of Z[x]? Is it a prime ideal?

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