= cos 4 + isin and B = cos + i sin. Prove that Bis not in Q(a).

Given: α=cos 2π7+i sin 2π7; β=cos 2π5+i sin 2π5To prove: β is not in Qα.

Answered: = cos 4 + isin and B = cos + i sin.…

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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Related questions

Question

= cos 4 + isin and B = cos + i sin. Prove that B
is not in Q(a).

Expert Solution

Step 1

Given: $α = \cos \frac{2 π}{7} + i \sin \frac{2 π}{7}; β = \cos \frac{2 π}{5} + i \sin \frac{2 π}{5}$
To prove: $β is not in Q (α) .$

Step 2

We have,
$α = \cos \frac{2 π}{7} + i \sin \frac{2 π}{7}; β = \cos \frac{2 π}{5} + i \sin \frac{2 π}{5}$
Now,
$α^{7} = {(\cos \frac{2 π}{7} + i \sin \frac{2 π}{7})}^{7} = \cos 7 (\frac{2 π}{7}) + i \sin 7 (\frac{2 π}{7}) = 1 We know that α \neq 1 . Similarly, β^{5} = (\cos \frac{2 π}{5} + i \sin \frac{2 π}{5}) = \cos 5 (\frac{2 π}{5}) + i \sin 5 (\frac{2 π}{5}) = 1 Also, β \neq 1 . α is a zero of the polynomial x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1 . β is a zero of the polynomial x^{4} + x^{3} + x^{2} + x + 1 .$