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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3
votes
1answer
107 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 ...
0
votes
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
votes
1answer
91 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 ...
0
votes
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
votes
0answers
45 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 ...
1
vote
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
votes
2answers
61 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 ...
0
votes
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}) ...
-2
votes
1answer
43 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
vote
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
votes
1answer
46 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 ...
1
vote
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.
1
vote
1answer
198 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 ...
1
vote
5answers
74 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
votes
1answer
39 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 ...
0
votes
1answer
39 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
79 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 ...
1
vote
0answers
61 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 ...
1
vote
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
votes
2answers
94 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
76 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
votes
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
votes
0answers
90 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
65 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
votes
1answer
215 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 = ...
-1
votes
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 ...
7
votes
1answer
214 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
vote
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
votes
1answer
53 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
votes
2answers
77 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
votes
1answer
67 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
105 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
87 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
117 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
votes
0answers
107 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
votes
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
149 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
136 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
77 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
304 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 $\mathrm{Tr}(A) = 0$. ...
-1
votes
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 $\alpha \ne 1$. ...
1
vote
3answers
47 views

summation and product of sin and cos

I wonder how to find summation for $\displaystyle \sum_{k=0}^{n-1}(\cos{\frac{2\pi k}{n}+i \sin\frac{2\pi k}{n}})$ and the same for product $\displaystyle \prod_{k=0}^{n-1}(cos{\frac{2\pi k}{n}+i ...
0
votes
1answer
104 views

Product of roots of unity

Does somebody have a nice proof of the following? $$\prod_{m=1}^{n-1} \frac{e^{2\pi i k m/n} - 1}{e^{2 \pi i m / n} - 1} = \begin{cases} 1 & \text{ if $\gcd(k, n) = 1$} \\ 0 & ...
8
votes
2answers
139 views

What is the value of $\sum_{m=1}^{19} \frac{1}{\zeta^{3m}+\zeta^{2m}+\zeta^{m}+1}$ with $\zeta=e^{2\pi i/19}$?

Given that $\zeta=e^{2\pi i/19}$, how to find the value of $$S=\sum_{m=1}^{19} \dfrac{1}{\zeta^{3m}+\zeta^{2m}+\zeta^{m}+1}$$? All I could think of was to somehow factorize the denominator and apply ...
1
vote
1answer
63 views

Question Primitive Roots of Unity

I am learning about primitive $n$-th roots of unity. I came across this statements while reading and was wondering why these were true: If $z$ is a primitive $n$-th root of unity and $n$ is even, ...
3
votes
5answers
129 views

How to solve $z^6+i=0$

I'm trying to solve $z^6+i=0$. I would have say that it's equivalent to $$z^6=-i\iff |z|^6e^{i6\arg(z)}=e^{i\frac{3\pi}{2}}\iff|z|^6=e^{i\left(\frac{3\pi}{2}-6\arg(z)\right)}$$ But I'm not able to ...
1
vote
0answers
81 views

$p$-adic $n$-th root of unity and $\exp(2\pi i /n)$

Let $n\geq 3$, and let $p$ be a prime number $\equiv 1 $ mod $n$. In complex numbers, we can write a primitive $n$-th root of unity as $\exp(2\pi i/n)$. Also, by Hensel's lemma, we see that $n$-th ...
2
votes
2answers
60 views

When is a sum of consecutive roots of unity an integer

Let $\xi \neq 1$ be an $n$th root of unity. When is a sum of the form $$ 1+\xi+\xi^2+\ldots+\xi^r, \quad 1 \leq r \leq n-1, $$ an integer? What are the possible integers? I suspect that the answers ...