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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

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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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1answer
140 views

Expressing the roots of $y^5 + 11y^4 - 77y^3 + 132y^2 - 77y + 11 = 0$ in terms of of $\zeta_{11}$?

Is there a concise way to express the roots of, $$x^3 + 7x^2 + 7x - 7 = 0$$ using the root of unity $\zeta_7$? Similarly, is there an analogous expression for the solvable quintic, $$y^5 + 11y^4 - ...
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57 views

$ S^2= \left(\sum_{j=0}^{p-1} \epsilon^{j^2}\right)^2$ where $p$ are prime number, $\epsilon $ is primitive p-th root of unity.

Let $p$ are prime number, $\epsilon $ is primitive p-th root of unity. Calculate: $$ S^2= \left(\sum_{j=0}^{p-1} \epsilon^{j^2}\right)^2$$ $p=3;p=5$ the result are real number. $\epsilon^{j^2}=\...
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1answer
352 views

Factorise $x^n+x^{n-1}+…+x+1$? [closed]

Is there a way to factorise $x^n+x^{n-1}+. . .+x+1$? I've tried to take the $1$ out, but now I do not know where to go from here because it doesn't seem to work.
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66 views

How can I calculate all results (3) of a cube root?

According to Wikipedia and Wolfram Alpha, a cube root $n^{\frac{1}{3}}$ has three results: one real number and two complex, if $n$ is a real number; and three complex numbers if $n$ is a complex. ...
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1answer
60 views

Extension of map $z\mapsto z^p$

Let $p$ be a prime and $G$ be the group of p-power roots of 1 in $\mathbb{C}$. Prove that $z\mapsto z^p$ is a surjective homomorphism and deduce that $G$ is isomorphic to a proper quotient of itself. ...
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83 views

To prove $\sum_{k=0}^{2016}\frac{1}{2017\zeta_{k}-1}=\frac{2017}{2017^{2017}-1}$

To prove $$\sum_{k=0}^{2016}\left(\zeta_k\prod_{~~~~~~j\neq k,\\ 0\leq j\leq 2016}(2017-\zeta_j)\right)=2017,$$ where $\zeta_0,\zeta_1,\cdots,\zeta_{2016}$ are the $2017$-th roots of the unity. ...
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1answer
31 views

Prove that $F(a_1, … , a_i)$ contains no $p$th roots of unity not in $F(a_1, … , a_{i-1})$

For context: each $a_i$ is a $p$th [$p=$prime] root of some element in the field $F(a_1,..., a_{i-1})$. The author claims that: "if $a_i$ is a $p$th root we can assume that $F(a_1, ... , a_i)$ ...
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147 views

Prove $\sum_{k=1}^{n}\sqrt{x^2+2x\cos\frac{(2k-1)\pi}{2n} + 1}\ge 1+ \sum_{k=1}^{n-1} \sqrt{x^2+2x\cos\frac{2k\pi}{2n} + 1}$

For nonnegative real number $x \ge 0$, and positive integer $n>0$,prove that $$ \sum_{k=1}^n\sqrt{x^2+2x\cos\frac{(2k-1)\pi}{2n} + 1} \ge 1+ \sum_{k=1}^{n-1} \sqrt{x^2+2x\cos\frac{2k\pi}{2n} + 1}$$ ...
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1answer
227 views

When is $\mathbb Z[\zeta_n]$ a Euclidean Domain?

After having accidentally duplicated this question, I thought I'd follow up with a related question. In an answer to the linked question, Zev Chonoles quotes the first page of Chapter 11 of ...
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141 views

When are the integers extended with the nth root of unity a unique factorization domain? [duplicate]

Let $\zeta_n$ be the $n$th root of unity. The wikipedia page for unique factorization domain (UFD) states that for $n \in \mathbb Z$, $1 \le n \le 22$, $\mathbb Z[\zeta_n]$ is a UFD, but not for $n = ...
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624 views

Vertices of a regular polygon in the complex plane

The problem given read as follows: Using complex numbers, find all the vertices of a regular polygon of $n$ sides knowing that its center is located at $z = 0$ and one of its vertices lies in $z_{1} =...
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1answer
102 views

2-power roots of unity in $\mathbb{Q}_p$

Let $p$ be an odd prime number. As a corollary of Hensel's lemma, we know that for $m\in \mathbb{N}$ there is a primitive $m$-th root of unity iff $m|(p-1)$. Hence, if $p\equiv 1$ mod 4, we have a $...
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1answer
54 views

How often is 2 a principal root of unity modulo a prime $p$?

We know, from Bertand's postulate that there is always a prime number between $n$ and $2n$, for $n > 3$. I'm wondering if there is a similar conjecture for the number $2$ being a principal root of ...
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1answer
116 views

Simple proof of existence of number field extension with the same roots of unity?

I'm looking for a brief proof of the following statement: Let $K$ be a number field and $p$ a prime number. Then there exists a field extension $L\supset K$ of degree $p$ with $\mu(L)=\mu(K)$. My ...
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4answers
311 views

Prove the nonzero eigenvalues of $M$ are roots of unity.

Suppose $A$ and M are $n \times n$ matrices over $\mathbb{C}%$, $A$ is invertible and $AMA^{-1} = M^2$. Prove the nonzero eigenvalues of $M$ are roots of unity. I get that you can rearrange this as $...
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3answers
402 views

How to express a power series in closed form

I am trying to express the power series $x + x^4/4! + x^7/7! + \cdots$ in closed form; I have already worked out the power series $1 + x^3/3! + x^6/6! + \cdots$ to be $(e^x + e^{x(2\pi i/3)} + e^{x(2\...
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2answers
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arctan series multisection by roots of unity

I'm trying to multisect the series for $\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} - \cdots$ using the method of roots of unity, as described in the paper linked in ...
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1answer
114 views

Simplifying Complex Numbers in Exponential Form [duplicate]

$(a)$ Suppose $p$ and $q$ are points on the unit circle such that the line through $p$ and $q$ intersects the real axis. Show that if $z$ is the point where this line intersects the real axis, then $z ...
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1answer
60 views

Explicit expressions for the $c_0,\ldots,c_{p-2}\in\Bbb{Z}$ satisfying $(1-\zeta_p)^{p-1}=p\sum_{k=0}^{p-2}c_k\zeta_p^k$.

It is well known that in the cyclotomic ring $\Bbb{Z}[\zeta_p]$ one has the equality of ideals $$(1-\zeta_p)^{p-1}=(p).$$ I'm struggling to find 'manageable' expressions for the associated units $$\...
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1answer
108 views

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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6answers
141 views

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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0answers
137 views

Linear Combination of Roots of Unity Equal to Zero?

Let $p$ be an odd prime number, and let $h\in\{1,\ldots,p-1\}$ be a primitive element of the multiplicative group $(\mathbb{Z}/p\mathbb{Z})^{\times}$, i.e., $\{h^{k}\,(\mathrm{mod}\,p):1\leq k\leq p-1\...
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About $n$'th rooth of $-1$ in a finite field

This question is based on this other question from @Malkoun, and more specifically the comments. Context. Let's recall the following proof which shows that $-1$ is a square if $\mathbb F_p$ for $p\...
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3answers
166 views

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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0answers
52 views

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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55 views

Can $\sqrt[3]{2}$ be expressed as a linear combination of roots of unity? [duplicate]

I need to solve the following problem: Can $\sqrt[3]{2}$ 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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1answer
42 views

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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2answers
87 views

Can $\sqrt{3}$ be written as a polynomial expression in $\sqrt[3]{3}$ and $\zeta_3$

I believe that it cannot be done. But now I can only think of the method using the basis of $\Bbb Q(\zeta_3,\sqrt[3]{3})$, which is brute force and tedious. So I am now searching for a rather simple ...
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153 views

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 ...
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3answers
311 views

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 $re^{i\theta}$. [4] In this question, I can work ...
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0answers
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Is there a term for a set of eigenvalues not containing any roots of unity except 1?

Is there a term describing a matrix $A \in \mathbb C^{n\times n}$ whose eigenvalues are all either $1$ or not a root of unity? Is there a term for a holomorphic selfmap $F$ of $\mathbb C^n$ fixing $0 ...
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2answers
34 views

Why there may not be a $n$-th primitive root in splitting fields of $x^n-1$ of positive characteristic?

I do not quite understand why there may not be a $n$-th primitive root in splitting fields of $x^n-1$ of characteristic non- $0$. May I please ask for the reason (I tried to understand it and it ...
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1answer
69 views

How can I see $\Phi_{p^r}(x)=(x^{p^{r-1}})^{p-1}+(x^{p^{r-1}})^{p-2}+\cdots+(x^{p^{r-1}})+1$ and is irreducible?

I am told that for a prime $p$, natural number $r$. $\Phi_{p^r}(x)=(x^{p^{r-1}})^{p-1}+(x^{p^{r-1}})^{p-2}+\cdots+(x^{p^{r-1}})+1$ and is irreducible (Here $\Phi$ denotes the cyclotomic polynomial). ...
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1answer
188 views

Find minimal polynomial for a period of 11th root of unity

Let $\alpha=\zeta_{11}+\zeta_{11}^3+\zeta_{11}^4+\zeta_{11}^5+\zeta_{11}^9$. Find the minimal polynomial $m_{\alpha,\mathbb{Q}}(x)$, and likewise for $\gamma=\zeta_{11}+\zeta_{11}^{-1} $. I started ...
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2answers
322 views

On the trigonometric roots of a cubic

We have, $$x^3+x^2-2x-1=0,\quad\quad x =\sum_{k=1}^{2}\,\exp\Bigl(\tfrac{2\pi\, i\, (6^k)}{7}\Bigr)\\ x^3+x^2-4x+1=0,\quad\quad x =\sum_{k=1}^{4}\,\exp\Bigl(\tfrac{2\pi\, i\, (5^k)}{13}\Bigr)\\ x^3+x^...
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81 views

When is a product of sums of roots of unity an integer?

A recent question asked for the residue modulo $101$ of $$A = \prod_{a=0}^9 \prod_{b=0}^{100} \prod_{c=0}^{100} (w^a + z^b + z^c)$$ where $w = e^{2\pi i/10}$ and $z = e^{2\pi i/101}$. I gave an answer ...
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1answer
103 views

Intermediate fields of $x^4+1$

Had this question on my test. We were asked to give irreducible polynomial for $e^\frac{2\pi i}{8}$ over $\mathbb{Q}$, which I said was $x^4+1$. We were asked to find the splitting field of this ...
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1answer
299 views

P-adic roots of unity appearing in expansion of root of polynomial

My question concerns the following: For a given polynomial $F(x) = a_n x^n + \cdots + a_0$ with a root $F(x) = 0 , x \in \Omega_p$, (that is the completion of the p-adic numbers) and $a_j \in Z_p$, ...
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1answer
152 views

Primitive root of unity in finite local rings

Let $p$ be a prime integer and $R$ be a finite local ring. Assume that $p||R^\times|$. Then by Cauchy's Theorem, there always exists a primitive $p$ root of unity in $R^\times$. Here $R^\times$ is a ...
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complex numbers; roots of unity

I have problem with these 2 exercices: 1) Multiply out and simplify $(a+bw)(a+bw^2)$ where $ω=e^{2πi/3}$ I only know that$(a+bw)(a+bw^2)= a^2+abw^2+abw+b^2$ and I don't know what to do next 2) If $...
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3answers
52 views

Let $A, B, C \in \mathbb{C}$, $n \in \mathbb{N}$. Prove that if $A^{n} = B^{n} = C^{n} = 1$ and $A + B + C = 0$, then $n$ is a multiple of 3.

I think the problem is clear from the title. It included a hint which suggested reducing it to the case $A = 1$, but so far I've come up with little. It is quite direct from the statement that $A, B, ...
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0answers
170 views

fifth roots of unity with integer coefficients

Considering the commutative ring Gamma 5, fifth roots of unity, with integer coefficients what can we say about elements which will be units? Are all possible values in the ring units, or are there ...
2
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0answers
77 views

Roots of Unity form a Real Algebraic Group

Does the group $\mu_n$ of $n^{\mathrm{th}}$-roots of unity form an algebraic group over $\mathbb{R}$ for every $n$? If so, why? I understand that $\mu_n$ forms an algebraic group over $\mathbb{C}$, ...
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3answers
156 views

Minimal polynomials of real algebraic multiples of roots of unity.

For roots of unity, the minimal polynomial is given by cyclotomic polynomials. Can we extend this to real algebraic multiples of roots of unity? For $r\omega_p$ I think there must be some way to ...
2
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2answers
434 views

sum of (some) $p$-th roots of unity, $p$ prime

Let $p$ be a prime and suppose $e_1,e_2,\cdots,e_p$ denote some $p$-th roots of unity in $\mathbb{C}$, not necessarily distinct. If $$e_1 + e_2 + \cdots + e_p=0,$$ then, can we always conclude that ...
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3answers
65 views

How can I give the solutions of $\ x^5-7$?

I'm trying to solve the equation $\ x^5-7$, which have complex solutions, how can I calculate the solutions of the equation? Thank you.
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1answer
150 views

Calculate product of $n$-th roots of unity

Consider $$a_k = \cos \frac{2k\pi}{n} -2 + i \sin \frac{2k\pi}{n}, \: n, k \in \mathbb N^*, n \text{ fixed}$$ I have to calculate $$\prod_{k=1}^n a_k = \prod_{k=1}^n [(\cos \frac{2k\pi}{n} + i \sin \...
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1answer
49 views

Gauss sum variation $\sum_{n=0}^{p-1}\left(\frac{a+bn}{p}\right)\zeta_p^{cn} = ?$

I'm having trouble evaluating this, for $a, b, p$ all pairwise coprime, $p$ an odd prime, $c$ any integer. $$\sum_{n=0}^{p-1}\left(\frac{a+bn}{p}\right)\zeta_p^{cn}$$ Any help/references would be ...
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1answer
119 views

Complex numbers involving roots of unity

Let $z\in \mathbb{C}$ and $n\in \mathbb{N}, n \geq 1$. Solve the following equation: $$z+z^2+\dots+z^n=n|z|^n$$ Obviously, $(z,n)=(0,n)$ and $(1,n)$ are solutions, $\forall n \geq 1$. Considering $z \...