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Roots of Unity

A root of unity is a complex number that, when raised to a positive integer power, results in 1. Roots of unity have applications to the geometry of regular polygons, group theory, and number theory.

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 \(a\).

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