# Tagged Questions

numbers $z$ such that $z^n=1$ for some natural number $n$; here usually $z$ is in $\mathbb C$ or some other field

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### Find all $z \in {\mathbb C}$ such that $z^{12}=1$ and $1+z+z^2+z^3+z^4+z^5 \in {\mathbb R}$

So, my first thought. If $z^{12} = 1, z \in$ is a twelfth root of unity. Knowing this, I can write $z = e^{i {2k \pi} \over {12}}$, with k $\in \{0,1,2,3,4,5,6,7,8,9,10,11\}$. Then if I just ...
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### If $k \in \mathbb{N}, n={2^k}$. Probe that $w$ is a primitive nth-root of unity $\iff$ $w$ is a root of $P_k = x^{2^{k-1}} + 1$

Going to the right is understandable. $w$ is a $2^k$-th primitive root of unity $\implies$ $w$ is a root of $P_k = x^{2^{k-1}}$ $w \in G_{2^k} \implies w^{2^{k}} = 1$ But then $w$ is a ...
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### $p \in Q[x]$ has as a root a fifth primitive root of unity, then every fifth primitive root of unity is a root of $p$.

I'm extremely stuck. Can't figure it. The conjugate is easy: let $w$ be a primitive root of unity, then $w^{-1}$ will also be a root, that's easy. But I'm missing $w^2$ and $w^3$. Why would they be ...
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### Complex roots for integer polynoms $p(z)=0$ with restriction $| z|=1$

There have been may questions about (integer) polynomials of sin and cos. There have been nearly as many ingenious answers using trigonomic-magic (converting sin, cos, tan into each other), but I (not ...
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### If $X$ has character $\chi$ and degree $d$. Prove that $g \in N$ if and only if $\chi(g) = d$ . Hint: Show that $\chi(g)$ is a sum of roots of unity.

Let $X$ be a matrix representation. And $N ={\{g \in G: X(g) = I}\}$. If $X$ has character $\chi$ and degree $d$. Prove that $g \in N$ if and only if $\chi(g) = d$ . Hint: Show that $\chi(g)$ is a ...
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### Is there a theoretical (or practical) definition of $n$-gon, for $n < 0$?

Background This is purely a "sate my curiosity" type question. I was thinking of building a piece of software for calculating missing properties of 2D geometric shapes given certain other properties,...
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### Finding fifth and tenth roots of unity in rectangular form.

Exercise: Find the fifth and tenth roots of unity in algebraic form. This is an early exercise in Ahlfors Complex Analysis. What I have tried so far: For the fifth roots I have tried reducing the ...
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### Why do I get an imaginary result for the cube root of a negative number?

I have a function that includes the phrase $(-x)^{1/3}$. It seems like this should always evaluate to $-(x^{1/3})$. For example, $-1 \cdot -1 \cdot -1 = -1$, so it seems that $(-1)^{1/3}$ should equal ...
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### What is $1^{1/n}$ for $n=\infty$ or for irrational $n$? [closed]

The $n$th root of $1$ produces a $n$ sided regular polygon centered around $0+0i$ with a vertex at $1+0i$. However, if we have $n=\infty$, then we should get an infinite sided polygon? Or more ...
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### On multiplying symmetric matrices by diagonal matrices with roots of unity

Given two symmetric non-zero and non-identity matrices $A,B\in\Bbb C^{n\times n}$ of same rank supposing there exists a non-identity diagonal matrix $D\in\Bbb C^{n\times n}$ containing only roots of ...
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### nth root of unity in a cyclic group $\mathbb{Z}_p^*$

Is there a specific set of steps that should be taken in order to the $n$-th root of unity in a cyclic group. To be more specific, I am trying to find the $8$th root of unity for $\mathbb{Z}_{17}^*$. ...
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### Finding the Roots of Unity

I have the following equation, $$z^5 = -16 + (16\sqrt 3)i$$I am asked to write down the 5th roots of unity and find all the roots for the above equation expressing each root in the form $re^{i\theta}$....
I have to do this: Find the value of $\sqrt[3]{\dfrac{1-i}{1+i}}$ in binomial and polar form. I have arrived to this point: z=\sqrt[3]{\dfrac{1-i}{1+i}}=\sqrt[3]{\dfrac{1-i}{1+i}\cdot\dfrac{1-...