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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

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.3: The Characteristic Of A Ring
Problem 18E
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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!
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 ($ - $²)² = -3 for any primitive third root of unity ( in Hint: use polynomial division to show that the primitive 3rd roots of unity are exactly the roots of x2 + x + 1, then apply this to simplify the expansion of (5 - 5)². (b) Show (55 + S3 – – )² = 5 for any primitive fifth root of unity 5 in (c) Show (C - 5)² = 2 for any primitive eighth root of unity ( in any field. any field. -4 -2 any field.
Transcribed Image Text:2) (a) Show ($ - $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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