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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Conjecture about some group semiring representations ( and roots of unity ).

Let $\Bbb R_+=[0,\infty)$ be a semiring. $\Bbb R_+[C_n]$ is the group semiring formed by the semiring $\Bbb R_+$ and the cylic group $C_n$. Let $\Bbb R_+[X_n]$ be the polynomial semiring. (...
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Conjecture about some rings and roots of unity. [duplicate]

Let $\Bbb R_{\geqslant 0}[X_n]$ be a polynomial semiring. More precisely $\Bbb R_{\geqslant 0}[X_n]$ are the polynomials of $X_n$ with positive real coefficients with $(X_n)^n = 1$. Let $F(n)$ be ...
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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, ...
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Simple-looking bound on root of unity

I am trying to prove some bound and stuck with the following: If $|n|\leq 3N/4$, then $\left|e^{2\pi in/N}-1\right|\geq\dfrac{n}{N}$ ($n,N$ are integers) How can I prove it?
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Conjecture about some group ring representations.

In this link : http://bandtechnology.com/PolySigned/ A set of numbers is described : $P(N)$. $ P(3),P(4),P(5),... $ are all (algebraicly closed) group rings. Identify $PN$ with $R[X_N]/(1+X_N+(X_N)^...
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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(\...
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1answer
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Adjoining two primitive n-th roots

Let $\omega_n$ denote a primitive $n^{th}$ root of unity. If $m$ and $n$ are positive integers with $lcm(m,n)=k$, show that $\mathbb{Q}(\omega_n,\omega_m)=\mathbb{Q}(\omega_k)$. To start, I am aware ...
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How find this value of $\prod_{1\le i<j\le n}(w^i-w^j)^2$

give the positive integer number $n$, and $w=\cos{\dfrac{2\pi}{n}}+i\sin{\dfrac{2\pi}{n}}$ where $i^2=-1$ find the vaule $$\prod_{1\le i<j\le n}(w^i-w^j)^2$$ My try:note $$w^n=1$$ $$\...
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$\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$,where $c \in \mathbb Z_9$ and $w=e^{2\pi i/9}$

$$\sum_{x \in \mathbb Z_9}w^{x^2-cx}=?$$ where $c \in \mathbb Z_9$, $w=e^{2\pi i/9}$ and $\mathbb Z_9$ is the ring of integers modulo 9.
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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 $\arg(\sum_{i=0}^t e^{2i\pi\frac{j_i}{q}} ) \in [0,\pi[$.
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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 ...
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A finite field extension $K\supset\mathbb Q$ contains finitely many roots of unity

I must show that any finite field extension $K\supset \mathbb Q$ can only contain finitely many roots of unity. I reasoned in the following manner: Let $n<\infty$ be the degree of $K$ over $\...
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117 views

Divisibility involving root of unity

Let $p$ be a prime number and $\omega$ be a $p$-th root of unity. Suppose $a_0,a_1, \dots, a_{p-1}, b_0, b_1, \dots, b_{p-1}$ be integers such that $a_0 \omega^0+a_1 \omega^1+ \dots a_{p-1} \omega^{p-...
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Quantum Fourier Transform and roots of unity.

I need to find $QFT_{6}$ for the state quantum state $\frac{1}{\sqrt2}(|0\rangle + |3\rangle)$. I received a very sufficient answer recently on simplifying nth roots of unity, but I am having a lot of ...
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How does one simplify exponents for complex primitive nth roots of unity?

Let us define a complex primitive N-th root of unity, omega: $$ \omega = \cos(\theta) + i\sin(\theta) \\ = e^{\frac{2\pi}{N}} $$ By the definition of an nth root of unity, ω is the second solution to ...
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Prove that $(x^2-x^3)(x^4-x) = \sqrt{5}$, where $x= \cos(2\pi/5)+i\sin(2\pi/5)$

Prove $(x^2-x^3)(x^4-x) = \sqrt{5}$ if $x= \cos(2\pi/5)+i\sin(2\pi/5)$. I have tried it by substituting $x = \exp(2i\pi/5)$ but it is getting complicated.
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How can one show that no cyclotomic field contains $\sqrt[3]{2}$?

This is a paraphrase of a problem from an old exam that I'm going over. Show that for all positive integers $t$, when $\omega$ is a primitive $t$-th root of unity, $\sqrt[3]{2}$ does not lie in the ...
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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 $...
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Zero sum of roots of unity decomposition

It's known that sum of all $n$'th roots of some $z \in \mathbb C$ with $|z| = 1$ is zero (if $n \geqslant 2$). Is it true that any zero sum of roots of unity can be decomposed in this way? That is if ...
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A property of power series and the q-th roots of unity

I'm trying to understand why if $ \displaystyle \sum_{n=0}^{\infty} a_{n}x^{n} = f(x) $, then $$ \sum_{n=0}^{\infty} a_{p+nq} x^{p+nq} = \frac{1}{q} \sum_{j=0}^{q-1} \omega^{-jp} f(\omega^{j} x)$$ ...
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For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative?

Let $k>0$ be an integer. For what $k$ is $P(k):=\prod_{j=1}^{13}\cos\frac{\pi kj}{13}$ negative? Since $13$ is prime, and for $\gcd(m,13)=1$, $P(2m)=P(2)=2^{-12}$ (can be shown by considering the ...
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1answer
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How do you calculate the kernel of the substitution homorphism $\mathbb{Z}[x] \rightarrow \mathbb{Z}[e^{i2\pi /r}]$?

I can do that for "easier" substitutions like $x \mapsto i$, but I don't know enough about primitive $rth$ roots yet. Does this require pieces of Galois theory? $r$ is a positive integer. I think ...
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Find the value of this logarithmic expression involving fifth root of unity.

Let $\alpha$ be the fifth root of unity. We then want to evaluate the expression $$\log |1 + \alpha + \alpha^2 + \alpha^3 - 1/\alpha |$$ Thanks in anticipation for your help in solving this!
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Looking at the intermediate fields of $\Bbb{Q}_7 = \Bbb{Q}(\omega)/\Bbb{Q}$ where $\omega = e^{i2\pi/7}$ .

From Basic Abstract Algebra (Robert Ash): The question that I'm concerned with is number 3, but I will write problems 1 and 2 as well, since they are all related... We now do a detailed analysis ...
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Question on roots of unity

This may seem absurd but what is wrong with the next reasoning about $n$th roots of unit?. For $k,l\in\mathbb Z$ such that $0 \leq k < l \leq n-1$: $$ e^{2\pi i k/n} = (e^{\pi i})^{2 k /n} \overset{...
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Why does Wolfram Alpha say that $\sqrt{1}=-1$? [duplicate]

Why does Wolfram Alpha say that $\sqrt{1}=-1$? Is this a mistake or what? Can anyone help? Thanks in advance.
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Is $\sqrt 7$ the sum of roots of unity?

Let $a_n$ and $b_n$ be 2 sequences of $n$ rationals. Is it possible that $\sqrt 7 = \sum_{m=1}^{n} a_m (-1)^{b_m}$ ? Is it possible that $\sqrt{17}$ = $\sum_{m=1}^{n} a_m (-1)^{b_m}$ ? How to ...
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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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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 $\mathbb{Z}^n$ ...
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Simple Question on Roots of Unity

The question asks: Find integers $p$ and $q$ such that $(p + qj)^{5} = 4 + 4j$ The question prior to this was: Find the fifth roots of $4 + 4j$ in the form $re^{j\theta }$, where $r > 0$ and $-\...
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Way to show $x^n + y^n = z^n$ factorises as $(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n$

For odd $n$ the Fermat equation $x^n + y^n = z^n$ factorises as $$(x + y)(x + \zeta y) \cdots (x + \zeta^{n-1}y) = z^n,$$ where $\zeta = e^{2 \pi i/n}$. I tried seeing this was true by multiplying ...
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Does $\lim_{n\to\infty}\sum\limits_{k=1}^n|e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi i(k-1)}{n}}|$ exist?

Does $\lim_{n\to\infty}\sum\limits_{k=1}^n|e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi i(k-1)}{n}}|$ exist? If yes, what is its value?
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Evaluation of an expression regarding a complex fifth root of unity

Let $\omega$ denote a complex fifth root of unity. Define $b_k = \sum_{j=0}^4j\omega^{-kj}$ for $0\le k\le 4$. What is the value of $\sum_{k=0}^4b_k\omega^{k}$?
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Can the expression be simplified?

If $1, a_1, a_2,\ldots, a_{n-1}$ are $n$-th roots of unity, can the following expression be simplified? $(1-a_1)(1-a_2)\cdots(1-a_{n-1})$?
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284 views

Primitive roots of unity in $\mathbb{Z}/p$

Can anyone help me with this problem? Let $p$ be a prime number. Prove that if the field $\mathbb{Z}/p$ has a primitive $n^{th}$ root of unity, then $n \mid (p-1).$ Any sources or books for ...
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Question on a homomorphism of a set G.

I'm having difficulty showing the given a map, say $\phi(z)=z^k$, is surjective. This question is from D & F section 1.6 - #19 Let $G$ =$\{z \in \mathbb C|z^n=1 \text{ for some } n \in \mathbb ...
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Why is the intersection of Q($\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|...
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A radical extension with a non-radical subextension

For a Galois Theory class I've been asked to find a radical extension with a non-radical subextension (all over $\mathbb{Q}$). So, I'm looking at the splitting field of $x^7 - 1$, namely $\mathbb{Q}(\...
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Why can't we just say 1 instead of “unity”?

I know this is a soft question of sorts but I am curious why we can't just say "1" instead of "unity," e.g. a root of unity.
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Radical extension

Let $K=\mathbb{Q}(\sqrt[n]a)$ where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d.$ Prove that $E=\mathbb{Q}(\sqrt[d]a)$. It's ...
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A Trigonometric Sum Related to Gauss Sums

This is a problem given to me by fractals on Art of Problem Solving. I couldn't solve it so I'm posting it here for some thoughts on it. Let $$S = \sum_{j = 0}^{\lfloor n/2 \rfloor} \sin\dfrac{2\pi\...
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The sum of the first and last $k$-th roots of unity

Let $\zeta_k$ be the first complex primitive $k$-th root of unity, i.e. $\zeta_k = e^{2\pi i/k}$. Then the last complex primitive $k$-th root of unity is given by $\zeta_k^{k-1} = \zeta_k^{-1} = e^{-2\...
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For which values of $k$, $0\leq k \leq n-1$, is $e^{i2πk/n}$ a primitive nth root of unity?

I know that the $n$-th root of unity is a primitive nth root of unity if, and only if, $k$ is relatively prime to $n$, but how do you prove it?
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Alternating Series Using Other Roots of Unity

$\sum (-1)^n b_n$ is representative of an alternating series. We look at whether $\sum b_n$ converges and if $b_{n+1}<b_n$ $\forall n\in \mathbb{Z}$. What if our alternating series is of the form $...
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Roots of unity in a field generated by a root of a polynomial

The polynomial x^3 − 2 has no roots in F7 and is therefore irreducible in F7[x]. Adjoin a root k to make the field F := F7(k), which will be of degree 3 over F7 and therefore of size 343. The ...
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show the rule between $w$, $w^2$, …$w^{n-1}$ with $w^n$ given w an nth root of unity

Let $w≠1$ be an $n$-th root of unity, i.e., $w^n-1=0$. Show that $1+2w+3w^2+\dots+nw^{n-1}=-\frac n{1-w}$. My question is how to relate $w, w^2, \dots,w^{n-1}$ with $w^n$?
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Fastest Way to Find order of element in Finite Fields?

Two questions: I use Miller-Rabin to find a prime, p, close to an arbitrary input number (this is very fast). Then I use Floyd's cycle finding algorithm to find the order of a randomly chosen element ...
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230 views

Cyclotomic Polynomials

Let $E(n)$ denote an nth root of unity. (For convenience, we may take $E(n) = \exp(\frac{2πi}{n})$.) Prove that for any prime $p$ and any natural number $r$, we have $$ \prod_{\substack{j\\ \gcd(p^r, ...
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1answer
114 views

Discriminant and roots of $ x^{n^2} \pm (x-1)^{n^2}$?

When considering the polynomials $x^{n^2} \pm (x-1)^{n^2}$ ( $n$ integer > 1 ) i noticed some things that appeared weird to me. Discriminant($x^{n^2} + (x-1)^{n^2}) = (n^2)^{n^2}$. Discriminant($x^{n^...
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3answers
889 views

3rd roots of unity as eigenvectors

Determine all eigenvalues of the matrix $$A=\begin{bmatrix}0&0&1\\1&0&0\\0&1&0\end{bmatrix}$$ and then determine a base for each eigenspace. It's easy to compute $\chi_A(z)...