Suppose f(x) E Z,[r] and f(x) is irreducible over Z,, where p is prime. (a) If deg f(x) = n, prove that Z,[r]/ (F(x)) is a field with p" elements. (b) Use the above problem to construct a field with 27 elements.
Suppose f(x) E Z,[r] and f(x) is irreducible over Z,, where p is prime. (a) If deg f(x) = n, prove that Z,[r]/ (F(x)) is a field with p" elements. (b) Use the above problem to construct a field with 27 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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![Suppose f(r) E Z,[r] and f(x) is irreducible over Z,, where p is prime.
(a) If deg f(x) = n, prove that Z,[r]/ (F(x)) is a field with p" elements.
(b) Use the above problem to construct a field with 27 elements.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5958936e-64d8-4f9f-ac16-af104f1eb32d%2F8bf36336-59dd-4d15-a8e1-c93049fbefe4%2F0n4l1ye_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose f(r) E Z,[r] and f(x) is irreducible over Z,, where p is prime.
(a) If deg f(x) = n, prove that Z,[r]/ (F(x)) is a field with p" elements.
(b) Use the above problem to construct a field with 27 elements.
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