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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7
votes
2answers
264 views
+500

Bounding a sum involving a $\Re((z\zeta)^N)$ term

This is a follow up to this question. Any help would be very much appreciated. Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$. Let ...
7
votes
1answer
191 views

Determining if a complex number is a root of unity

How would you determine if $a+ib$ is an nth root of unity? Obviously, the modulus of $a+ib$ must be $1$. But you would also need to determine if $a+ib$ is located at a vertex of a regular ...
2
votes
1answer
47 views

Show that $\cot \frac{\pi}{2m}\cot \frac{2\pi}{2m}\cot \frac{3\pi}{2m}…\cot \frac{(m-1)\pi}{2m}=1$

Prove: $$\cot \frac{\pi}{2m}\cot \frac{2\pi}{2m}\cot \frac{3\pi}{2m}...\cot \frac{(m-1)\pi}{2m}=1$$ This is a roots of unity problem. I managed to show a similar example for $\cos$ by the ...
0
votes
0answers
12 views

Complex Number Powers of Coprime Rational Powers

I'm trying to figure out $z^{p/q}$ where $p,q$ are coprime. Suppose I want to find $z^{2/7}$ where $z=128$. I can rewrite $z=128e^{0}$ Now I know that the $z^{1/7}$ roots are $2e^{k2\pi i/7}$ for ...
-1
votes
0answers
38 views

$A,B\in\mathbb Q[x]$ with $A,B$ monic, and $ AB\in\mathbb Z[x]$, prove $A,B\in\mathbb Z[x]$

It is part of cyclotomic polynomials. But I don't know how to deal with it and what to do next. I have prove $n$-th root is related to Euler's totient fuction. But I don't know how to use it. Thank ...
4
votes
0answers
78 views

Finite Messy Trigonometric Sum

Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$ The source of this ...
4
votes
1answer
65 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 ...
0
votes
1answer
19 views

Proof that $t^m=t^{j}$ if $t$ is an $r^{th}$ root of unity such that $r \mid k$.

I need help with the following proof. Let $j$ = $0,1,\ldots, k-1$. Also, let $t$ be an $r$th root of unity other than $t=1$ such that $r \mid k$. We know $m=j\pmod k$. Furthermore, $m$, $j$ and $k$ ...
0
votes
1answer
45 views

Show that $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ and $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}))\cong \mathbb{Z_n}^*$ [duplicate]

Question 1: For $n \in \mathbb{N}$ explain why $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ We must show that it is normal and separable. $\mathbb{Q}(\zeta_n)/\mathbb{Q}$ is normal since it ...
0
votes
1answer
60 views

How many fields are there between $\mathbb{Q}$ and $\mathbb{Q}(\zeta_{15})$?

$\mathbb{Q}(\zeta_{15})$ is a field extension of $\mathbb{Q}$, where $\zeta_{15}$.I am trying to find the number of $i$ such that: $\mathbb{Q} \subset L_i \subset\mathbb{Q}(\zeta_{15})$ Is there a ...
2
votes
2answers
71 views

Find $ [\mathbb{Q(\alpha)}: \mathbb{Q}] $ where $ \alpha=\zeta+\zeta^3+\zeta^4+\zeta^5+\zeta^9 $

Suppose $ \zeta$ is a primitive $ 11$-th root of unity and $ \alpha=\zeta+\zeta^3+\zeta^4+\zeta^5+\zeta^9 $ Find $ [\mathbb{Q(\alpha)}: \mathbb{Q}] $ Could someone please give me a hint ...
0
votes
3answers
29 views

Sum of n-th roots of unity [duplicate]

I'm being asked to prove that $$1 + \omega + \omega^2 + ... + \omega^{n-1} = 0$$ where $\omega \ne 1$ is an n-th root of unity, and I don't know where to start I feel like there's something terribly ...
1
vote
3answers
96 views

How to solve $x^3 = 1$?

My intuitive side tells me to take the cube root of both the sides and get the answer $1$. However, I realize that it might be a problem for I'll lose solutions as given here: Is it the case that ...
2
votes
1answer
40 views

Denominator is product of irreducibles with cyclic Galois group

Short version of the question: Guess the next terms in the sequence : $D_{17},D_{19},D_{23}$ etc where $$ \begin{array}{lcl} D_3 &=& (a\pm 1) \\ D_5 &=& (a\pm 1) (a^2-1 \pm 11a) \\ ...
3
votes
0answers
32 views

Proving that a Galois group is cyclic

Let $K$ be a field containing a primitive $n$th root of unity and let $F = K(t)$ be the field of rational functions over $K$. I'm having trouble proving that for each $n > 1$ the field $F$ is Galois ...
2
votes
2answers
142 views

Simplifying this (perhaps) real expression containing roots of unity

Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ although I don't think that is relevant. Let $\zeta:=\exp(2\pi i/k)$ and $\alpha_v:=\zeta^v+\zeta^{-v}+\zeta^{-1}$. ...
25
votes
3answers
5k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
18
votes
6answers
7k views

Intuitive understanding of why the sum of nth roots of unity is $0$

Wikipedia says that it is intuitively obvious that the sum of $n$th roots of unity is $0$. To me it seems more obvious when considering the fact that $\displaystyle ...
1
vote
0answers
28 views

Roots of unity in fixed field of decomposition group.

$\zeta_q\in L^{G({\frak P})}\Leftrightarrow$ if $\sigma\in$ $G(\frak p$) then $\sigma(\zeta_q)=\zeta_q\Leftrightarrow$ if $\sigma$ fixes $\frak P$ then $\sigma$ fixes $\zeta_q$. What's next? Suppose ...
0
votes
0answers
16 views

Norm of roots of unity in a subextension.

Suppose $L/k$ is a Galois extension of number fields, $G$ is the Galois group, and for $\frak p$ a prime ideal of $\cal O$$_L$, let $K=L^{G(\frak p)}$, where [$L$ : $K$] = $p$. Suppose $m$ is the ...
0
votes
1answer
11 views

$f(z_0)=\sum_{n=0}^{k-1}z_0^n + 1+1+ \dots$ - Dense on the unit circle

Personal question : Let $z_0$ be a $2^k$th root of unity. We obtain for the function $f(z)=\sum_{n \geq 0} z^{2^n}$ (radius of convergence $R=1$) that $f(z_0)=\sum_{n=0}^{k-1}z_0^n + 1+1+ \dots$. Why ...
0
votes
1answer
24 views

The norm and the angle of the complex number $\sqrt[3]{-1}\sqrt[6]{7}$?

I am learning the roots of unity here. I want to express arbitrary complex number in radical form in terms of the norm and the angle to use the formula $e^{i\phi}=\cos(\phi)+i\sin(\phi)$. Consider ...
29
votes
1answer
344 views

A spiralling sequence based on integer divisors. Has anyone noticed this before?

Firstly, please excuse the informal style of my explanation, as I am not a mathematician, although I am aware that this can be explained in more formal terms. I have mapped integers to points on a ...
0
votes
1answer
52 views

Solve $z^4 = 1$ for all $z$ , Hence or otherwise, solve $z^4 =(z-1)^4$

Question: Solve $z^4 = 1$ for all $z$. Hence or otherwise, solve $z^4 =(z-1)^4$. My attempt for the first part $$z^4 = 1$$ $$z^4 -1 =0$$ $$ (z^2-1)(z^2+1) = 0 $$ $$ ...
0
votes
1answer
33 views

I want to find the 6th root of z which lies in a specific domain.

Let $z=-3+8.5i$, $\arg z \in (-\pi,\pi]$. Find the $6th$ root of $z$ which which lies in: $(2\pi\frac{4}{6},2\pi\frac{5}{6})$ Provide an answer to decimal places. What is $\theta$ if the answer ...
1
vote
1answer
43 views

sum of roots of unity multiplied by k+1

I'm considering the following sum: $$\sum\limits_{k=0}^n (k+1)\epsilon^k,$$ where $\epsilon=e^{\frac{2\pi i}{n}}$. I write the sum as $$\frac{\rm d}{{\rm d}\epsilon}\sum\limits_{k=0}^n ...
2
votes
1answer
24 views

Invertibility of a certain matrix attached to a primitive root of unity.

Let $q\in \mathbb{C}$ be a primitive $n$th root of unity, for some $n>1$. Consider the $n^2\times n^2$-matrix $$M=\left( q^{ki+lj}\right)_{(k,l),(i,j)}$$ indexed by all pairs $(k,l), (i,j) \in ...
2
votes
3answers
68 views

Solve for $z$ in $z^3=8i$

I only have a question about the end results. I answered the question fully but my professor knocked off 1 point for my $w_1^0$ result, but I don't know why. He circled the $i\pi /6$ in my answer but ...
1
vote
4answers
105 views

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 ...
4
votes
3answers
62 views

Alternating sum of roots of unity $\sum_{k=0}^{n-1}(-1)^k\omega^k$

Consider the roots of unity of $z^n = 1$, say $1, \omega, \ldots, \omega^{n-1}$ where $\omega = e^{i\frac{2\pi}n}$. It is a well known result that $\sum_{k=0}^{n-1}\omega^k = 0$, but what if we want ...
-1
votes
1answer
50 views

Let $\zeta_n$ be the $n^{th}$ root of unity $\zeta_n=e^{2\pi i/n}$. How can I prove that $\zeta_5\notin \mathbb{Q}(\zeta_7)$?

This question is from Artin 15.3.3: Let $\zeta_n$ be the $n^{th}$ root of unity $\zeta_n=e^{2\pi i/n}$. How can I prove that $\zeta_5\notin \mathbb{Q}(\zeta_7)$? I'm quite stuck so any help would be ...
1
vote
1answer
46 views

roots of unity and cyclotomic polynomials over $\mathbb{F}_p$

Given a prime $p$, Let $n = p^d -1$ and let $f$ be an irreducible polynomial dividing the $n$-th cyclotomic polynomial in $\mathbb{F}_p[t]$. Let $\alpha = t + (f)$ in $\mathbb{F}_p[t]/(f)$ where $(f)$ ...
0
votes
0answers
31 views

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 ...
1
vote
5answers
56 views

$ 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 ...
1
vote
1answer
33 views

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 ...
1
vote
1answer
32 views

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 ...
1
vote
3answers
80 views

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 ...
0
votes
2answers
56 views

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 ...
5
votes
1answer
109 views

Finding $\sqrt{17}$ and $\sqrt{257}$ in the regular $17$-gon and $257$-gon?

(Edit: I need to revise this question with my original intent. Pls do not answer it yet. Thanks.) Given the regular $n$-gon formed by the $n$-th roots of unity. For some $n$, how do we find ...
3
votes
1answer
70 views

Show that the elements of the form $1+\zeta + \zeta^2 + \dots + \zeta^m$, with $1 \leq m \leq p-2$, are invertible in $\mathbb{Z}[\zeta]$

Let $\zeta = e^\frac{2 \pi i}{p}$, with $p$ prime. Show that the elements of the form $1+\zeta +\zeta^2 + \dots + \zeta^m$, with $1 \leq m \leq p-2$, are invertible in $\mathbb{Z}[\zeta]$. I know ...
0
votes
3answers
41 views

Linear Combination of Roots of Unity

Let $\omega_n$ be a primitive $n$the root of unity and $\lambda_k$ be natural numbers. Does $\sum_{k=1}^{n} \lambda_k w_n^k =0$ imply $\lambda_1 = \lambda_2 = ... = \lambda_n $? I am aware of ...
0
votes
1answer
24 views

Why is the sum of the first $k$ powers of a $k$-th primitive root $\varphi_k$ of $1$ always $0$?

Let $\varphi_k\in\mathbb{C}$ be a primitive root of $1$. It turns out, that $$\varphi_k^1+\ldots+\varphi_k^k=0\text{ .}$$ If I draw the roots for some fixed $k$, I can see that this seems evident. For ...
5
votes
2answers
69 views

Find complex roots of $\frac{2i}{1+i}$

Find 6th roots of $$\frac{2i}{1+i}$$ $$\frac{2i}{1+i}=\frac{2e^{i\pi/2}}{\sqrt 2 e^{i \pi/4}}=\sqrt 2 e^{i \pi/4}$$ Now if I set $z^{1/6}=\sqrt 2 e^{i \pi/4}$ and knowing the fact that the roots ...
-2
votes
2answers
139 views

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 ...
3
votes
4answers
80 views

All Solutions for $(-256)^{\frac{1}{4}}$ and $1^{\frac{1}{5}}$ Imaginary Roots?

This is a question about the imaginary roots of the two equations $$ (-256)^{\frac{1}{4}} \qquad\text{and}\qquad 1^{\frac{1}{5}}. $$ For the first one I've worked out that 2 of the solutions are ...
4
votes
2answers
75 views

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 ...
0
votes
1answer
32 views

What is the intersection of $\mathbb Z$ with the ideal generated by $1-\zeta_n$?

For example, 1-(-1) is in the ideal <2>, whenever $n$ is even. Suppose $R=\Bbb Z[\zeta_n]$, and (by the above), that $n$ is odd. We know 1+$\zeta_n$ can be multiplied by ...
0
votes
2answers
30 views

Why in a field of characteristic $p$, $\zeta_p \sqrt[p]t$ is not a root of $X^p-t\in \mathbb F_p(t)[X]$

I know that in a field such that $Car(K)=p$ is prime, the $X^p-t\in \mathbb F_p(t)[X]$ has a unique root (I know how to prove it and thus, it's not the question). But in the usual logic, $X^p=t\iff ...
3
votes
1answer
55 views

What is the sum of ALL of the nth powers of the qth roots of unity?

I am reading a Wikipedia article on Ramanujan's sum. The article states that: It follows from the identity $x^q − 1 = (x − 1)(x^{q−1} + x^{q−2} + ... + x + 1)$ that the sum of the nth powers of All ...
1
vote
0answers
29 views

How can I choose $\frak p\unlhd\cal O$ prime so $u\in\cal O^\times$ becomes a $n$-th power (mod $\frak p$)?

$k$ is an algebraic number field, and $\cal O$ is the ring of integers, $\cal O^\times$ is the set of invertible elements of $\cal O$. Suppose $u\in\cal O^\times$ is not a $n$-th power. How can I ...