1. We proved thta the ideal < x + 1 > is prime in Z[x] by definition. Inthis problem, show this ideal is prime, but not maximal, in Z[x] using theFHT as follows:(a) Define the substitution homomorphism μ-1: Z[x] → Z by μ-1 (p(x)) =p(-1) - you can assume this is a homomorphism. Show it is onto(b) Prove keru_1 =< x + 1 >(c) Then apply the FHT to answer the question(d) Now use the FHT again, to explain why < x + 1 > is a maximl idealof F[x],where F is any field

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Answered: 1. We proved thta the ideal < x + 1 >…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.4: Maximal Ideals (optional)

Problem 26E: . a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show...

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