Problem 1. Consider the polynomial ring Z-[X]. (a) Show that the polynomial P:= X² +4 is an irreducible element of Z,[X). (b) Show that the quotient ring k defined below is a field: k:= Z[X]/(X² +4). What is the cardinality of k? (c) Are the rings k and Z49 isomorphic? Justify your answer.
Problem 1. Consider the polynomial ring Z-[X]. (a) Show that the polynomial P:= X² +4 is an irreducible element of Z,[X). (b) Show that the quotient ring k defined below is a field: k:= Z[X]/(X² +4). What is the cardinality of k? (c) Are the rings k and Z49 isomorphic? Justify your answer.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![Problem 1. Consider the polynomial ring Z-[X].
(a) Show that the polynomial P:= X² +4 is an irreducible element of Z[X].
(b) Show that the quotient ring k defined below is a field:
k:= Z[X]/(X² +4).
What is the cardinality of k?
(c) Are the rings k and Z49 isomorphic? Justify your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7068779e-a624-45ba-8c15-d1ec8879533a%2Fc7f716b0-1296-4bb6-8bb3-0c76c17bbc5e%2Fs4gxyzf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 1. Consider the polynomial ring Z-[X].
(a) Show that the polynomial P:= X² +4 is an irreducible element of Z[X].
(b) Show that the quotient ring k defined below is a field:
k:= Z[X]/(X² +4).
What is the cardinality of k?
(c) Are the rings k and Z49 isomorphic? Justify your answer.
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