7. Suppose R is an integral domain having the property that Rac] is a PID. Prove that R is a field as follows: (a) Explain why the substitution homomorphism o: R[x] →→R (that is, Ho (p(z)) = p(0)) is onto (b) Determine kerpo as a principal ideal (just write what it equals - you don't have to prove it). Why must kerjo be prime? (c) Now, what can we say about a non-zero prime ideal in a PID? (d) Make the conclusion that R is a field
7. Suppose R is an integral domain having the property that Rac] is a PID. Prove that R is a field as follows: (a) Explain why the substitution homomorphism o: R[x] →→R (that is, Ho (p(z)) = p(0)) is onto (b) Determine kerpo as a principal ideal (just write what it equals - you don't have to prove it). Why must kerjo be prime? (c) Now, what can we say about a non-zero prime ideal in a PID? (d) Make the conclusion that R is a field
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![7. Suppose \( R \) is an integral domain having the property that \( R[x] \) is a PID. Prove that \( R \) is a field as follows:
(a) Explain why the substitution homomorphism \( \mu_0 : R[x] \rightarrow R \) (that is, \( \mu_0(p(x)) = p(0) \)) is onto.
(b) Determine \( \ker \mu_0 \) as a principal ideal (just write what it equals - you don’t have to prove it). Why must \( \ker \mu_0 \) be prime?
(c) Now, what can we say about a non-zero prime ideal in a PID?
(d) Make the conclusion that \( R \) is a field.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F905809b3-f86f-4a48-848e-914423a7f462%2F0537f55f-5b78-4ec9-9c13-b204ba5fe01d%2Fubvw56g_processed.jpeg&w=3840&q=75)
Transcribed Image Text:7. Suppose \( R \) is an integral domain having the property that \( R[x] \) is a PID. Prove that \( R \) is a field as follows:
(a) Explain why the substitution homomorphism \( \mu_0 : R[x] \rightarrow R \) (that is, \( \mu_0(p(x)) = p(0) \)) is onto.
(b) Determine \( \ker \mu_0 \) as a principal ideal (just write what it equals - you don’t have to prove it). Why must \( \ker \mu_0 \) be prime?
(c) Now, what can we say about a non-zero prime ideal in a PID?
(d) Make the conclusion that \( R \) is a field.
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