n nth root of unity is primitive if 5ª ± 1 for any proper c the following, Euler's identity e" = cos(0) + i sin(0) may
n nth root of unity is primitive if 5ª ± 1 for any proper c the following, Euler's identity e" = cos(0) + i sin(0) may
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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--Need help with this problem from Number Theory--
![An nth root of unity in a field K is an element (e K such that C" = 1.
An nth root of unity is primitive if 5ª # 1 for any proper divisor d of n.
i0
In the following, Euler's identity e" = cos(0) + i sin(0) may be helpful!](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7140e3e-b877-4eb0-97a8-e066e434192f%2F91d69d51-0f85-4b1c-9943-1cea83271764%2F6yarie4_processed.png&w=3840&q=75)
Transcribed Image Text:An nth root of unity in a field K is an element (e K such that C" = 1.
An nth root of unity is primitive if 5ª # 1 for any proper divisor d of n.
i0
In the following, Euler's identity e" = cos(0) + i sin(0) may be helpful!
![2)
(a) Show ($ - $²)² = -3 for any primitive third root of unity ( in
Hint: use polynomial division to show that the primitive 3rd roots of unity are exactly
the roots of x2 + x + 1, then apply this to simplify the expansion of (5 - 5)².
(b) Show (55 + S3 – – )² = 5 for any primitive fifth root of unity 5 in
(c) Show (C - 5)² = 2 for any primitive eighth root of unity ( in any field.
any
field.
-4
-2
any
field.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff7140e3e-b877-4eb0-97a8-e066e434192f%2F91d69d51-0f85-4b1c-9943-1cea83271764%2Fqnn17wi_processed.png&w=3840&q=75)
Transcribed Image Text:2)
(a) Show ($ - $²)² = -3 for any primitive third root of unity ( in
Hint: use polynomial division to show that the primitive 3rd roots of unity are exactly
the roots of x2 + x + 1, then apply this to simplify the expansion of (5 - 5)².
(b) Show (55 + S3 – – )² = 5 for any primitive fifth root of unity 5 in
(c) Show (C - 5)² = 2 for any primitive eighth root of unity ( in any field.
any
field.
-4
-2
any
field.
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