Roots of Unity

Roots of Unity Warmup


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

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

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 \(a + bi\) for real numbers \(a\) and \(b\) is \(a\).


Problem Loading...

Note Loading...

Set Loading...