a.) To find the 9th roots of unity:
We have,
,
,
[by De-Moivre's law]
,
[by De-Moivre's law]
Let,
,
Thus, the 9 roots of unity are given by
i.e.
1,
,
,
,
,
,
,
,
b.) To find the 6th roots of unity:
We have,
[by De-Moivre's law]
Thus, the 6 roots of unity are given by
i.e.
i.e.
NOTE:
According to the De-Moivre's formula all nth roots of a complex
number
are given by
, k=0,1,2,...,n-1
List and graph the 6th roots of unity. What are the generators of this group? What are the primitive 6th roots of unity?
find the 6th roots of -144
what is roots of unity and Primitivity
30. Roots of Polynomials. Find the roots of the following polynomials, using the complex exponential and roots of unity where necessary: (a) z4422 4 = 0 (b) 24422+ 16 = 0 (c*) (zi5-(z - i)5 = 0 (d) 2432 z1 = 0
30. Roots of Polynomials. Find the roots of the following polynomials, using the complex exponential and roots of unity where necessary: (a) z4422 4 = 0 (b) 24422+ 16 = 0 (c*) (zi5-(z - i)5 = 0 (d)...
Find the cube roots of -64 i. Graph each cube root as a vector in the complex plane. Choose the correct cube roots below. O A. 64cos 60° + i sin 60º), 64(cos 180° + i sin 180), 64(cos 300° + i sin 300°) OB. 4cos 30° + i sin 300), 4(cos 150° + i sin 150), 4(cos 270° + i sin 270º) OC. 4cos 90° + i sin 90), 4(cos 210° + i sin 210), 4(cos 330° + i...
Find the cube roots of -64 i. Graph each cube root as a vector in the complex plane. Choose the correct cube roots below. O A. 64(cos 60° + i sin 60%), 64(cos 180° + i sin 180°), 64(cos 300° + i sin 300°) OB. 4(cos 30° + i sin 300), 4(cos 150° + i sin 150°), 4(cos 270° + i sin 270º) OC. 4(cos 90° + i sin 90°), 4(cos 210° + i sin 210), 4(cos 330° + i...
fekri/n k0,1,...,n-1}, called the nth roots of unity. A primitive root of unity is = eri/n for which 2. The roots off(x) = x"-1 are the n complex numbers Cn and are ged(n, k) 1. It is easy to see that Q(C) is the splitting field of zn - 1. (a) For each n 3,... ,8, sketch the nth roots of unity in the complex plane. Use a different set of axes for each n. Next to each root, write...
3. Supposeis a primitive 12th-root of unity. Then show that ç5,57 and are also primitive 12th-roots of unity
29. Roots and Factors. For each of the following find the roots of the given equations and sketch the roots in the complex plane: (a) cube roots z3 = 1 (b) square roots z2 = i (c) sixth roots z6 = -64 (d) fifth roots z5 = 32e5Ti/3
Let
and
, the subgroup of fourth roots of unity.
(a) Characterize all left cosets of H.
(b) Prove or disprove that G/H isomorphic to G.
G =C H={+1, i)