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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4
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1answer
71 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 ...
2
votes
1answer
32 views

Considerations on cyclotomic extensions

I stopped in front of some issues regarding certain passages of the theorem 9.4 of Algebra of Serge Lang, in which we suppose to have k a field such that $[k (\mu_n):k]=\phi (n)$ where $\mu_n$ is ...
2
votes
1answer
111 views

Find the minimum polynomial of a sum of roots of unity.

Let $ \omega $ be an 11-th primitive root of 1 over $ \Bbb Q $ Let $ \beta = \omega + \omega^9 $ Find $ [ \Bbb Q ( \beta) : \Bbb Q ) ] $ and Find the minimum polynomail of $\beta$. I asked a ...
2
votes
1answer
83 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 $x^n=1$ ?
1
vote
1answer
34 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 ...
5
votes
0answers
67 views

Singularities at roots of unity

I want to construct a function $f$ with the following properties: $f$ has a singularity at $z=1$, and for any $\zeta = e^{2\pi i\frac{a}{b}}$ with $(a,b)=1$, then $$\lim\limits_{x\to1^-}\frac{f(\zeta ...
5
votes
0answers
64 views

Geometrical construction of 7th roots of unity given parabola x^2

Given parabola $x^2$ on plane, how can I construct 7th roots of unity? I was straying with my only idea, that sum of squares of real and imaginary part of roots of 1 equals 1 and belongs to $\mathbb{...
4
votes
0answers
82 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
0answers
111 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 ...
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 ...
3
votes
0answers
73 views

Does roots of L-polynomial of curve being roots of unity imply the curve is supersingular?

If $C$ is a supersingular curve over $\mathbb F_q$, then $\frac1{\sqrt q}$ of roots of $L$ function are roots of unity. What about converse? If $\frac{1}{\sqrt q}$ of roots of $L$ polynomial are ...
3
votes
0answers
35 views

Proving unit of quartic number field is fundamental

Let $K = \mathbb Q(\alpha)$, for $\alpha$ a root of $a^4 + 4 \alpha^2 + 2 = 0$. I want to prove the group of units $\mathcal O_K^*$ equals $\langle -1, \alpha^2 + 1\rangle$. I've found the ring of ...
3
votes
0answers
100 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 $\mathbb{...
3
votes
0answers
119 views

Irrational roots of unity?

Is it possible to take irrational roots of unity? For example, say I wanted to solve $f(x)=(x+1)^{\sqrt{2}}=1$. I found that one solution is the obvious $x=0$, and another one can be written nicely as ...
3
votes
0answers
112 views

Finite Trigonometric Sum

I have a dynamical system model whose equilibria depend on the solution of the following finite sum: \begin{align} \sum_{j\neq i}^n\frac{\sin(\theta_j-\theta_i)}{\left(1-\cos(\theta_j-\theta_i)\right)...
3
votes
0answers
66 views

For which $m$ is this sum of roots of unity $0$?

A professor at my university gave me the following problem: For what real values of $m$ do we have $$L=\lim_{N\to\infty}\frac{1}{N}\sum_{k=1}^Ne^{2\pi ik^3m}=0$$ If $m$ is rational, expressed ...
3
votes
0answers
115 views

root of a unit in a real biquadratic field

Let $p_1$ and $p_2$ two primes numbers $\equiv 1\pmod 4$. If we note by $\varepsilon_m$ the fundamental unit of real quadratic field $\mathbb{Q}(\sqrt m)$, then how can be solved in $\mathbb{Q}(\...
2
votes
0answers
38 views

How many tubes can you balance in a centrifuge?

I recently learned that if you have a centrifuge whose number of holes $n$ is divisible by $6$, then you can balance any number of tubes except for $1$ and $n-1$. If $k$, the number of tubes you want ...
2
votes
0answers
272 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.
2
votes
0answers
103 views

Textbook says roots of unity is equal to 1

The elements of the set $U_n = \{z \in \mathbb{C} : Z^n =1 \}$ are called the $n^{\text{th}}$ roots of unity. Using the technique of Examples 1.6 and 1.7, we see that the elements of this set are the ...
2
votes
0answers
39 views

What primes can ramify and decompose in $k(\mu_{p^m}) \mid k$?

Let $k$ be a number field and $p$ be a rational prime. Then consider the extension $k(\mu_{p^m}) \mid k$ of adjoining all roots of unity of degree $p^m$ to $k$. Assuming that $\mu_{p^m} \cap k$ is ...
1
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0answers
30 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 ...
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 ...
1
vote
0answers
22 views

If a, b, and c are real, and w is a non real cube root of unity…

If a, b, c, and d are real numbers and w is a non-real cube root of unity, and if 1/(a + w) + 1/(b + w) + 1/(c + w) + 1/(d + w) = 2/w Prove that 1/(a + w^2) + 1/(b + w^2) + 1/(c + w^2) + 1/(d + w^2)...
1
vote
0answers
32 views

Is there always a zero of this polynomial that is not a root of unity?

Consider the polynomial $1-z(1+z)^a$ with $a$ a positive integer. Is there always a complex zero of this polynomial that is not a root of unity? I tried to prove it by induction or by contradiction ...
1
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0answers
26 views

$p$-divisibility and $q$-th roots of unity

Let $p$ be a prime, $n$ a positive integer and $q=p^n$ a $p$- power. Let $w,v$ is a root of the $q,p$-th cyclotomic polynomial $Q_q,Q_p$, respectively. Let $f$ be a polynomial with coefficient in $\...
1
vote
0answers
72 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
0answers
122 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 ...
1
vote
0answers
119 views

What is the condition for roots of conjugate reciprocal polynomials to be on the unit circle?

Given an Nth order complex polynomial $P(z) = \sum\limits_{n=0}^N a_nz^n$ such that $a_n = a^*_{N-n}$ i.e. conjugate reciprocal, Lakatos and Losonczi mention that a necessary and sufficient condition ...
1
vote
0answers
74 views

How to quickly find a root of unity in a ring?

Lets say we're in a field where multiplication and addition are modded against some prime number P (so it's defined for {0,....,P-1} Lets fix a number N < P, such that a root of unity can be found....
1
vote
0answers
79 views

About the rank of (sub) matrices whose entries are roots of unity

Let $\Omega$ be a matrix with entries $a_{jk}=\omega^{jk}$, where $0\leq j,k\leq n-1$, and $\omega=e^{-2\pi i/N}$, with $N\in \mathbb{N}$, so $\Omega$ looks like $$ \Omega=\begin{pmatrix} 1 & 1 &...
1
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0answers
121 views

A question on primitive root of unity

Let n be a positive integer and let $\alpha$ , $\beta$ be primitive n-th roots of unity. a) Show that $\frac{1-\alpha}{1-\beta}$ is an algebraic integer. b) If $n\geq 6$ is divisible by at least two ...
1
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0answers
72 views

May someone tell me where I went wrong when finding a closed-form solution to a recursion?

Let $r(n)=\left\lceil\frac{n}{4}\right\rceil$, where $k \in \mathbb{N}$. Note that this can be rewritten as a recursion $R(n)=R(n-4)+1$, where, $R(0)=0, R(1)=1, R(2)=1, R(3)=1$. Since this recursion ...
1
vote
0answers
73 views

Roots of unity in CM-field

Let $K$ be a CM-field, ie. a totally imaginary quadratic extension of a totally real number field $F$ and let $p > 2$ be a rational prime. My question simply is Are the $p$-th roots of unity, ...
1
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0answers
74 views

Understanding a.. weird definition

I came across the following definitons: Let $n$ be a positive integer and $\zeta_n$ be a primitive $n^{th}$ root of unity. For any prime $p\nmid n$, let the degree of the extension $\mathbb{Q}_p(\...
1
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0answers
31 views

argument of a lacunary sum of roots of unity

Let $q>4$ and $t< \sqrt{q}$ be integers. Determine the set $\{j_1,...,j_t\}$ of integers $0 \leq j_i <q-1$ such that $\arg(\sum_{i=0}^t e^{2i\pi\frac{j_i}{q}} ) \in [0,\pi[$.
1
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0answers
190 views

sum of $p$th roots of unity

Let $p$ be an odd prime. Let $\mathbb Z_p$ be the ring of integers from $0$ to $p-1$, i.e. $\mathbb Z_p=\left\{0,1,\dots,p-1\right\}$. Let $r$ be a positive integer. Then, for all values of $p$, is ...
1
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0answers
133 views

Simplifying a product over roots of unity

Let $\zeta_{n}=e^{2\pi i /n}$ be the nth root of unity. Now consider the product : $$\prod_{k=1}^{n-1} (1-\zeta_{n}^{k})^{\zeta_{n}^{k}}$$ Is there a simple formula for this product as a function of $...
1
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0answers
79 views

prove that polynomial has root of unity

Prove that $ f=x^n\pm x^m\pm1 $ is either irreducible over rationals or has a root which is a of unity. I tried to see what if $x=|r|e^{i\phi}$ but I have no proper result.
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0answers
223 views

Evaluating a polynomial at a root of unity?

Let $R = \mathbb{Z}[x]/(x^n+1)$ be the $2n$th cyclotomic ring (for $n$ a power of $2$ in which case $\Phi_{2n}(x) = x^n+1$). Let $g$ be an $n$-dimensional vector chosen at random from $\mathbb{Z}^n$ ...
1
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0answers
156 views

Why is the intersection of Q($\sqrt[n]{a}$) and Q(nth root of unity) Galois? (a>0, n an integer)

With this, I can show the intersection is either Q or Q($\sqrt{a}$). All I have is that their intersection must be real and all subfields of Q($\sqrt[n]{a}$) are of the form Q($\sqrt[d]{a}$) where d|...
1
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0answers
291 views

How does trigonometry in a Galois field work?

This is a follow-up to this question. I'm interested in doing trigonometry in finite fields on a computer. I do not understand precisely how trigonometric functions are supposed to work in a finite (...
0
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0answers
21 views

Trace of roots of unity has valuation more than 1

Let $F$ be a finite extension of $\mathbb{Q}_p$ (p is prime) and $K/F$ be a unramified extension of degree $\ell\ge 3$, where $\ell$ is a prime and not equal to $ p$. Denote $\mu_K$ be the group of ...
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 $...
0
votes
0answers
19 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
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 ...
0
votes
0answers
29 views

$(ab^{n+1})^{1/n}$ where $a,b \in \mathbb{C}$

Let $a,b \in \mathbb{C}$ and $n \in \mathbb{N}$. We can present $a$ and $b$ in polar form as \begin{equation} a = r_a \mathrm{e}^{i \theta_a} \quad \textrm{and} \quad b = r_b \mathrm{e}^{i \theta_b}, ...
0
votes
0answers
98 views

Carmichael function and primitive roots of unity

I have been reading about the Carmichael function recently and I would like to ask about some elementary implication of its properties as I haven't found it stated explicitly. If I understand it ...
0
votes
0answers
62 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 ...
0
votes
0answers
59 views

Isomorphism betwixt a Galois Group and the set of $n$th roots of unity

Let $p \in \mathbb{P}$ and $n \in \mathbb{N}: p \nmid n$; the set of $n$th roots of unity be $W_n$; $\mathbb{F}$ be a field$: char(\mathbb{F}) \in \left\{ {0,p}\right\}, \mathbb{F} \supseteq W_n$ (set ...