# Tagged Questions

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### Complex Number Geometry

Problem: There exist two complex numbers $c$, say $c_1$ and $c_2$, so that $2+2i$, $5+i$, and $c$ form the vertices of an equilateral triangle. Find the product $c_1c_2$. I've been struggling with ...
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### Why is the reciprocal of an $n$-th root of unity its complex conjugate?

As stated in the Wikipedia article on roots of unity, the reciprocal of an $n$-th root of unity is its complex conjugate. They provide the following proof of this statement: Let $z\in\mathbb{C}$ be a ...
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### Summing complex numbers of magnitude $1$

A professor at my university gave me the following problem: For what real values of $m$ do we have $$L=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^Ne^{2\pi ik^3m}=0$$ If $m$ is rational, expressed ...
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### Galois Group of $x^n - a$

Homework problem: If the field F contains a primitive nth root of unity, prove that the Galois group of $x^n - a$, for $a \in F$, is abelian. I'm not really sure where to start here and I'm ...
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### Problem understanding solution of complex nth-root of unity

a while ago we had the solution for a complex number task about the nth-root of unity in the complex. But now I am having some difficulties to fully understand it: The task was to find all complex ...
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### Roots of Unity - Complex Numbers

The sets $A = \{z : z^{18} = 1\}$and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A \ \text{and} \ w \in B\}$ is also a set of complex roots of unity. ...
Find the roots of $6z^5 + 15z^4 + 20z^3 + 15z^2 + 6z + 1 = 0.$ Erm...it sorta looks like $(x+y)^6 = x^6 +6x^5y + 15x^4y^2 + 20x^3y^3 + 15y^4x^2 + 6y^5x + y^6$ Any idea how to proceed from here? ...