Algebra

Roots of Unity

Roots of Unity Warmup

         

How many non-real solutions are there to the equation x7=1? x^7 = 1?

What is the product of the fourth roots of unity?

How many of the 12th roots of unity are not 4th roots of unity?

Let (φ1,φ2,φ3)=(1,1+3i2,13i2).(\varphi_1, \varphi_2, \varphi_3) = \left( 1, \frac{-1 + \sqrt{3}i}{2}, \frac{-1 - \sqrt{3}i}{2} \right). In other words, φ1,φ2,φ3\varphi_1, \varphi_2, \varphi_3 are the third roots of unity. Further suppose that you have a polynomial for some integer n>4,n > 4, f(x)=a0+a1x+a2x2+a3x3++a3nx3n.f(x) = a_0 + a_1x + a_2x^2 + a_3x^3 + \ldots + a_{3n}x^{3n}. Which of the following is equal to f(φ1)+f(φ2)+f(φ3)3?\frac{f(\varphi_1) + f(\varphi_2) + f(\varphi_3)}{3}?

How many of the 5th roots of unity have positive real part?

Note: The real part of a complex number that can be written as a+bia + bi for real numbers aa and bb is aa.

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