8. Give a counterexample to the claim of problem 7 when unique factorisation fails in Z[a].
8. Give a counterexample to the claim of problem 7 when unique factorisation fails in Z[a].
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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please answer question 8 thanks !!
![7. Let a be a root of X2-bX+c, for b, c integers such that b² - 4c < 0. Suppose
that unique factorisation holds in Z[a]. Let m, n be relatively prime positive integers
such that mn is expressible as x² + bxy + cy² for integers x and y. Show that both
m and n are expressible in this form.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87351664-192b-44e9-8681-fefec31a9dea%2Fa394fbed-a348-414f-8ffa-c7e1fdf56bb7%2Fxl19daa_processed.png&w=3840&q=75)
Transcribed Image Text:7. Let a be a root of X2-bX+c, for b, c integers such that b² - 4c < 0. Suppose
that unique factorisation holds in Z[a]. Let m, n be relatively prime positive integers
such that mn is expressible as x² + bxy + cy² for integers x and y. Show that both
m and n are expressible in this form.
![8. Give a counterexample to the claim of problem 7 when unique factorisation fails
in Z[a].](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F87351664-192b-44e9-8681-fefec31a9dea%2Fa394fbed-a348-414f-8ffa-c7e1fdf56bb7%2Fqg2oyhk_processed.png&w=3840&q=75)
Transcribed Image Text:8. Give a counterexample to the claim of problem 7 when unique factorisation fails
in Z[a].
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