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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34 views

Carmichael numbers and primitive roots of unity

Let $n$ be a Carmichael number. Is it possible for an integer ring $\mathbb{Z}_n$ to contain primitive $(n-1)^{th}$ roots of unity? Or do only only primitive roots of unity of degree $\quad k < ...
1
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1answer
35 views

Prove: there is a unique pair of integer roots of unity which differ in real part by $1$.

I saw the following lemma somewhere, and I hope I did not misread it: If $z_1$ and $z_2$ are $n$th and $m$th roots of unity respectively ($n,m$ positive integers possible equal), and the real part of ...
0
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0answers
36 views

Fast Fourier Transform: How is the roots of unity matrix divided?

For an example for input size N=8, how is the roots of unity matrix divided for a divide and conquer approach? My understanding is that it's divided into four quadrants, Ma with J&K evens; Mb ...
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3answers
76 views

Finding units in cyclotomic fields

I want to classify the six units in $\mathbb Z[\zeta_3]$, where $\zeta_3$ is a primitive cube root of unity. I know the basic idea of this is to show that the norm of $\alpha \in \mathbb Z[\zeta_3]$ ...
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2answers
53 views

Requirement on Norm for units in Cyclotomic Fields

Consider $\zeta_p$ a primitive $p^{th}$ root of unity. Prove that $\alpha \in \mathbb Z[\zeta_p]$ is a unit iff $N(\alpha)=\pm1$. I'm not even sure where to start with this. I know that ...
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5answers
122 views

Why is $\sqrt [n] 1$ not an expression “in radicals” of a root of unity?

In Edwards' Galois Theory, in the chapter on Cyclotomic polynomials, the author devotes a lot of effort to proving that prime order primitive roots of unity can be expressed "by radicals", and gives ...
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1answer
34 views

for $p$ given, $\zeta_p$ a primitive root of unity, fow which $d\in \mathbb{Z}$ does $\zeta_p \in \mathbb{Q}(\sqrt{d})$?

Here is a question that I am trying to answer: Let $p$ be a prime greater than $2$. For which $d \in \mathbb{Z}$ contains $\mathbb{Q}(\sqrt{d})$ a primitive root of power $p$? What I did If ...
3
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1answer
116 views

A Polygon is inscribed in a circle $\Gamma$

A regular polygon P is inscribed in a circle $\Gamma$. Let A, B, and C, be three consecutive vertices on the polygon P, and let M be a point on the arc AC of $\Gamma$ that does not contain B. Prove ...
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1answer
47 views

To prove: For $p$ prime, $\zeta \in \mathbb{C}*$ and $ord(\zeta) = p$, $(1-\zeta)$ is a prime ideal in $\mathbb{Z}[\zeta]$

I'm trying to solve the following: Let $p$ be a prime number and $\zeta \in \mathbb{C}^* $ such that $ord(\zeta) = p$. Show that $(1-\zeta)$ is a prime ideal in $\mathbb{Z}[\zeta]$ and that there is ...
3
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1answer
94 views

To prove: $ [K : \mathbb{Q}] = 2 \ \Longrightarrow \ \exists \zeta \text{ primitive root of unity}, \ \mathbb{Q}(\zeta) \ \supseteq \ K $

I have to show that the following statement is true: Let $K$ Be a field extending $\mathbb{Q}$ such that $[K: \mathbb{Q}] \ = \ 2$. Then there is a root of unity $\zeta$ such that $K \subseteq ...
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0answers
58 views

Roots of unit in a DVR

Let $A$ be a DVR such that its fraction field $K$ is complete w.r.t to the natural absolute value in $K$. I am trying to prove that the projection from $A$ to the residue class field, $F$, maps the ...
3
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0answers
46 views

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$?

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$? (I'm asking this question in order to understand this answer). My idea for a proof so far: $f:=X^{15}-t\in F_5(t)$ is the ...
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2answers
28 views

Are all roots of unity (solutions to $a^k \equiv 1$) for a prime modulo $p$, a multiple of $p-1$??

If $a$ is coprime to $p$, and $a \not \equiv 1,-1 \mod p $ then are there any solutions to $a^k \equiv 1 \mod p $ such that $0< k < p-1$? For any counterexample, it is obvious $GCD(p-1, k) \not ...
3
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2answers
62 views

Roots of $z^r=1,r\notin\mathbb{Q}$

If $a,b\in\mathbb{Z}$, and $\frac a b$ is in lowest terms, then $$z^{\frac a b}=1\\\implies z=\exp\left(\frac{2\pi in b}{a}\right)\forall n\in\mathbb{Z}$$ This means that $z$ has exactly $a$ distinct ...
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2answers
61 views

Find roots of polynomial

Let p be an odd prime number and $ζ =ζ_p = cos(2π/p)+isin(2π/p)$ How do you find all of the roots of the polynomial $f(x) = x^{p-1} +x^{p-2}+…+x+1$ How do you show that $p = (1-ζ)(1-ζ^2)…(1-ζ^{p-1}) ...
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1answer
45 views

Finding the elements of a number C based off of 24th and 54th roots of unity

Let $A$ be a set of all complex numbers $z$ such that $z^24=1$ and let $B$ be the set of all complex numbers $w$ such that $w^54=1.$ That is: \begin{align*}A&=\{z\;|\;z^{24}=1\}\\ ...
1
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1answer
25 views

Rings and equations

Let $R$ be a commutative (non-zero) ring with identity, what are the solutions of $x^2-1=0$? Obviously, $x=\pm 1$ are solutions and if $R$ is an integral domain there aren't other solutions since ...
2
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1answer
48 views

Cyclic properties of multiplicative group G of all the complex $2^n$ roots of unity

Consider the multiplicative group G of all the complex $2^n$ roots of unity, $n=0,1,2,\ldots$ I am asked to verify whether $G$ is a cyclic group and whether it has a finite set of generators. The ...
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1answer
26 views

Rational number in $\mathbb{Z}[\omega]$ should be integer.

Let $\omega = \cos \frac{2\pi}{p} + i \sin \frac{2\pi}{p}$ for some prime number $p > 2$. Then how to prove that if $q \in \mathbb{Q} \cap \mathbb{Z}[\omega]$, $q$ must be integer.
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1answer
204 views

Roots of unity polynomial [duplicate]

Let $\omega=e^{\frac{2\pi i}{n}}$. Prove that $\Pi_{k=1}^{n-1} (1-\omega^k)=n.$ So far, I've tried brute-forcing it by expanding out the product, but it ended up getting too messy--and now I'm ...
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5answers
76 views

How do you call this fact about sum of powers of n-th unity root?

I often see identity $$\sum_{k=0}^{n-1}e^{\tau ika/n} = \cases {n \quad \text{ if }n | a\\0\quad \text{ otherwise}}$$ in the context of generating functions. It allows to zero out all members of ...
2
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1answer
41 views

Sum of a proper subset of the $p^\text{th}$ roots of unity

I know, because I have read it, that if $p$ is prime, no sum of a proper subset of the $p^{\text{th}}$ roots of unity (in $\mathbb{C}$) is zero. I thought I knew how to prove this, but found to my ...
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1answer
40 views

How are sinusoids and roots of unity related to each other?

The discrete Fourier transform (DFT) is often teached as being a transform that decomposes a given signal or sequence of numbers into sinusoids with frequencies $\large\frac{k}{N}$ where $k \in [0, ...
2
votes
2answers
81 views

An $n \times n$ matrix with rational entries such that $A^{n+1}=I$

I'm working on finding $A \in M_n(\mathbb{Q})$ such that $A^{n+1}=I$. If $n$ is odd, $A=-I$ satisfies the condition. When $n$ is even, clearly it should have eigenvalues $e^{2 \pi ik/(n+1)}(k=1,\cdots ...
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0answers
66 views

Why $\sqrt{5}$ doesn't lie in $\mathbb{Q(\eta_{5})}$?

In Lorenz's Galois Theory book, there's a problem : Why $\sqrt{15} \notin \mathbb{Q(\eta_{15})}$, where $\eta_{15}$ is a $15$-th primitive root of unity ? But My question is about what it's ...
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2answers
100 views

Degree of the extension $\mathbb{Q}(\zeta_3,\zeta_7)$ over $\mathbb{Q}$

Question is to compute the degree of the extension $\mathbb{Q}(\zeta_3,\zeta_7)$ over $\mathbb{Q}$. We have ...
4
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2answers
96 views

is $\sqrt p$ in $\mathbb Q(\zeta_{4p})$?

i think for every prime $p$ we have $\sqrt p \in \mathbb Q(\zeta_{4p})$ when $\zeta_{4p}$ is a primitive 4p-th root of unity.but i have no idea to prove it. is it true? can any one help me with a ...
2
votes
2answers
77 views

Rationalizing the denominator in general

How do you rationalize the denominator of something like $$\frac{1}{\sqrt[n]{a_1}+\sqrt[n]{a_2}+...+\sqrt[n]{a_n}}$$? I'm thinking roots of unity.
0
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1answer
46 views

Primitive root of unity in finite fields

To find a primitive $n$-th root of unity in a field $F_q$ of size $q$, one takes the smallest positive integer $m$ such that $q^m \equiv 1 \bmod n$ and finds a primitive $n$-th root of unity in ...
4
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0answers
92 views

More on primes $p=u^2+27v^2$ and roots of unity

Given, $$p=u^2+27v^2=3m+1\tag1$$ and the cubic, $$x^3+x^2-mx+N=0\tag2$$ with its constant expressed in terms of $(1)$ as, $$N = \frac{1}{27}(1-3p\pm2pu)\tag3$$ and the sign $\pm u$ chosen ...
2
votes
2answers
71 views

Polynomial with positive coefficients

Consider a polynomial $P(x) = \sum_{i=1}^{n}{a_ix^{i-1}}$ in $\mathbb{C}$. Is it true that if $\{a_i\}$ are positive and not all equal, then $P(\exp(\frac{2i\pi}{n})) \neq 0$ ? Thanks
7
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1answer
216 views

What's so special about primes $x^2+27y^2 = 31,43, 109, 157,\dots$ for cubics?

While trying to find a closed-form solution for particular cubics as sums of cosines (related to this question), I came across this family with all roots real, $$F(x) = x^3+x^2-mx+N = ...
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3answers
57 views

Describe the solutions of the equation in terms of roots of unity?

I want to find the solutions of the equation $$\left[z- \left( 4+\frac{1}{2}i\right)\right]^k = 1 $$ in terms of roots of unity. When I try to solve this, I get \begin{align*}z - 4 - \dfrac i2 ...
8
votes
1answer
220 views

A little more on $\sqrt[3]{\cos\bigl(\tfrac{2\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{4\pi}7\bigr)}+\sqrt[3]{\cos\bigl(\tfrac{8\pi}7\bigr)}$

Using a special case of an identity by Ramanujan, we find that given the roots $x_i$ of $$x^3 + x^2 - (3 n^2 + n)x + n^3=0\tag1$$ which, since its discriminant is negative, always has three real ...
1
vote
1answer
30 views

Cyclotomic polynomial,

Show that $\displaystyle X^n-1=\prod_{d\mid n}\Phi_d(X)$. We have that $$\Phi_n(X)=\prod_{\underset{\gcd(i,n)=1}{1\leq i\leq n}}(X-\zeta_n^i)$$ where $\zeta_n=e^{\frac{2i\pi}{n}}$ therefore, we ...
1
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1answer
50 views

Given three vectors involving trigonometric functions, how many $\theta$ satisfy a particular box product relation?

If $$\vec a =(1+\sin \theta )\hat i+\cos \theta \hat{ j}+\sin2\theta\hat k\\ \vec b =(\sin( \theta +2\pi/3))\hat i+\cos ( \theta +2\pi/3) \hat{ j}+\sin( 2\theta +4\pi/3)\hat k\\ \vec c =(\sin ( \theta ...
2
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1answer
56 views

Radical expression for roots of unity

Can somebody point out a reference to the nested radical formula of the complex roots of unity when $n = 2^N$, i.e. in solving $x^n=1$ ?
3
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2answers
78 views

Determine whether ${\dfrac{2+i}{2-i}}$ is a root of unity

I need to determine whether ${\dfrac{2+i}{2-i}}$ is a root of unity. At first, I expressed this number as ${\dfrac{3}{5}+\dfrac{4}{5}i}$. Then I tried to use a formula for $\sin{nx}$, where x = ...
0
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1answer
69 views

Find all the roots of this complex equation

Let $C$ be the set of complex numbers and $j$ the imaginary unit. Find all the roots(in $z$ $\in$ $C$) of the following equation: $$ 2z^7 + 6z^4 = z^3e^{-j{\frac π7}} + 3e^{-j{\frac π7}} $$ ...
4
votes
1answer
110 views

$\sum_{\zeta^p=1}(\zeta-1)^n$

Given $n\geq0$ let $$ z_n=\sum_{\zeta^p=1}(\zeta-1)^n $$ where $p$ is an odd prime number (summation extended to all $p$-th roots of 1). It is clear that: $z_n\in\Bbb Z$ (it's a Galois invariant sum ...
3
votes
0answers
89 views

Automorphism(Galois groups) and galois theory

I've been stuck on two last parts for two different questions, can someone please help me with these. The first question is: Let $\sigma\in Aut(L/\mathbb{Q})$, where $L$ is some subfield of ...
3
votes
2answers
124 views

Finding the Extension Degree of a Cyclotomic Field

Greetings Mathematics Community. I am having much difficulty in solving the following problem: If $m\equiv 2$ (mod 4), show that $\mathbb{Q(\zeta_m)}=\mathbb{Q(\zeta_{\frac{m}{2}})}$ where $\zeta$ ...
2
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0answers
123 views

The sum of finite exponential series with a quadratic phase

How can I prove that: $$ \sqrt \frac K2 + i \sqrt \frac K2=\sum^K_{m=1}\exp\left(i\frac \pi Km^2\right) $$ When $K$ is even.
0
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1answer
45 views

Approximating a power of a root of unity to within $\delta$

I have an estimate of $\omega$, a root of unity. I'm really wondering how small the error (in the estimate), which I give as $\epsilon$, has to be, so that when I take my estimate of omega to the ...
4
votes
2answers
156 views

Nth root of Unity

Hi all I am in higher level mathematics and I am taking the IB. We started doing problems associated with nth root of unity. I understand how to find the roots of for example: $$Z^3 - 1 =0$$ and ...
3
votes
4answers
137 views

Norm of an element in cyclotomic extension (Exercises VI.19 Lang's Algebra)

Let $\zeta$ be a primitive $n^{\rm{th}}$ root of unity. Let $K=\mathbb{Q}(\zeta)$. If $n=p^r (r\geq 1)$ is a prime power, show that $N_{K/F}(1-\zeta)=p$ If $n$ is divisible by at least two distinct ...
0
votes
2answers
83 views

Show that complex numbers are vertices of equilateral triangle

1)Show if $|z_1|=|z_2|=|z_3|=1$ and $z_1+z_2+z_3=0$ then $z_1,z_2,z_3$ are vertices of equilateral triangle inscribed in a circle of radius. I thought I can take use from roots of unity here, since ...
0
votes
3answers
29 views

Are conditions equaivalent that they are roots of unity?

If I have conditions that $|z_1|=|z_2|=|z_3|=|z_4|=1$ and $z_1+z_2+z_3+z_4=0$ Is it suffice to state they are roots of unity ?
2
votes
3answers
317 views

Eigenvalues and roots of unity

Let $A \in \mathcal{M}_{n}(\mathbb{C})$ such that $A^{n} = \mathrm{I}_{n}$ and the family $(\mathrm{I}_{n},\ldots,A^{n-1})$ is linearly independent. I would like to prove that $\mathrm{Tr}(A) = 0$. ...
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2answers
81 views

How can I find fifth root of unity?

I have no idea to do this question, how can I find the fifth root of unity? Question : Find all the distinct fifth root of unity. Let $\alpha$ be a fifth root of unity such that $\alpha \ne 1$. ...