1. In the ring R=Z [√2], let M = {a+b√2|2|a}. Answer the following (a) Show M is an ideal of R (b) Show M is a maximal ideal of R, directly, using the definition of maximal ideal = (c) Now show M is maximal by showing F RM (that is, the quotuent ring R mod M) is a field as follows: Just show that F has only two elements (note √2 € M ) and hence must be the field of two elements
1. In the ring R=Z [√2], let M = {a+b√2|2|a}. Answer the following (a) Show M is an ideal of R (b) Show M is a maximal ideal of R, directly, using the definition of maximal ideal = (c) Now show M is maximal by showing F RM (that is, the quotuent ring R mod M) is a field as follows: Just show that F has only two elements (note √2 € M ) and hence must be the field of two elements
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
![1. In the ring R =Z [√2], let M = {a+b√2|2|a}. Answer the following
(a) Show M is an ideal of R
(b) Show M is a maximal ideal of R, directly, using the definition of
maximal ideal
(c) Now show M is maximal by showing F RM (that is, the quotuent
ring R mod M) is a field as follows: Just show that F has only two
elements (note √2 € M) and hence must be the field of two elements](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2F702b8965-1909-45bc-8f39-e13a39c83e2f%2Fkh5z9y_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. In the ring R =Z [√2], let M = {a+b√2|2|a}. Answer the following
(a) Show M is an ideal of R
(b) Show M is a maximal ideal of R, directly, using the definition of
maximal ideal
(c) Now show M is maximal by showing F RM (that is, the quotuent
ring R mod M) is a field as follows: Just show that F has only two
elements (note √2 € M) and hence must be the field of two elements
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