Questions tagged [roots-of-unity]

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

655 questions
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How to prove that $\cos(\frac{\pi}{13})$ is algebraic and not rational?

How to prove that $\cos(\frac{\pi}{13})$ is algebraic and not rational? My Try: $\cos(\frac{\pi}{13})+i\sin(\frac{\pi}{13})$ and $\cos(\frac{\pi}{13})-i\sin(\frac{\pi}{13})$ are roots of $x^{26} -1$....
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The sum of unity roots

Let $p\not=2$ be a prime. Suppose that $\zeta_1,\cdots,\zeta_n$ are $p$-th unity roots. If their sum $S$ is an integer, show that $S$ is congruent to $n$ modulo $p$. I don't know how to deal with ...
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There is a field $\Bbb F$ such that the equation $x^2=1$ have more than two solutions (for some $x\in\Bbb F$)?

There is a field $\Bbb F$ such that the equation $x^2=1$ have more than two solutions (for some $x\in\Bbb F$)? This question comes suddenly to my mind. I know that if $\Bbb F=\Bbb C$ then the ...
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Prove or disprove $\frac{1}{5}(3-4i)$ is a root of unity. [duplicate]

Prove or disprove $\frac{1}{5}(3-4i)$ is a root of unity. Here is the definition of root of unity: An nth root of unity, where $n$ is a positive integer $(i.e. n = 1, 2, 3, …)$, is a number $z$ ...
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Evaluating a series of rational functions by hand or using generating functions

I'll jump right into the question: Let $\omega$ be a primitive $n$th root of unity. Show that \begin{align*} \frac{1}{n} \sum_{i=0}^{n-1} \frac{1}{(1-\omega^i t)(1 - \omega^{-i}t)} = \frac{1-t^{...
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Can $\sqrt{2}$ be expressed as a linear combination of roots of unity? [duplicate]

I need to solve the following problem: Can $\sqrt{2}$ be expressed as a linear combination of roots of unity (with coefficients in $\Bbb Q$)? I am completely lost because I have no idea of how to ...
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How many ﬁelds are there strictly between $\Bbb Q(\zeta)$ and $\Bbb Q(\zeta^3)$

Let $\zeta$ denotes the $12$th primitive root of unity. I want to know that how many ﬁelds are there strictly between $\Bbb Q(\zeta)$ and $\Bbb Q(\zeta^3)$ and what are they. I think an obvious one ...
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Can $\sqrt{3}$ be written as a polynomial expression in $\sqrt{3}$ and $\zeta_3$

I believe that it cannot be done. But now I can only think of the method using the basis of $\Bbb Q(\zeta_3,\sqrt{3})$, which is brute force and tedious. So I am now searching for a rather simple ...
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Finding the roots of the polynomial $z^{12} = (z+2)^{12}$, where $z$ is a complex number, by using the $12$ roots of unity.

The method goes like this : $\displaystyle\frac{(z+2)^{12}}{z^{12}} = 1$ Let $\displaystyle w = \frac{z+2}z$ $w^{12} = 1$ The solutions to this equation are the $12$ roots of unity. But one ...
Solving $z^5=-16+16\sqrt3i$ using the fifth roots of unity
1 Write down the fifth roots of unity. Hence, or otherwise, find all the roots of the equation $$z^5=-16+16\sqrt3i,$$ giving each root in the form $re^{i\theta}$.  In this question, I can work ...