Algebra

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 $a$.

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