# Questions tagged [roots-of-unity]

The roots-of-unity tag has no usage guidance.

52
questions

**1**

vote

**1**answer

208 views

### Möbius inversion formula and roots of unity

Is the exact value of
$$
\sum_{d\mid n} \mu\left(\frac{n}{d}\right) \zeta^d
$$
known? Here, $\mu$ denotes the Möbius function and $\zeta$ a root of unity. At first sight, it seems to me that this ...

**8**

votes

**0**answers

193 views

### Computing coefficients of polynomials from roots in $O(n\log{n})$ time

Suppose I have a univariate polynomial $p$ over a prime-order finite field $\mathbb{F}_q$ whose roots I know.
Suppose that the roots of $p$ are always an $n$-sized subset of $R=\{1,2,\dots,N\}, N <...

**2**

votes

**0**answers

186 views

### Finite sum involving root of unity

I have the following sum:
$$\sum_{\sigma=1,\text{odd}}^{\frac{p}{2}-2}\frac{\sin \frac{r \sigma \pi}{p}}{\sin \frac{q \sigma \pi}{p}}$$
where $p\equiv2\pmod4$, $p$ and $q$ are coprime numbers such ...

**5**

votes

**0**answers

257 views

### Is an algebraic number satisfying certain super-congruences a root of unity?

Let $K|\mathbb{Q}$ be a number field, $D$ its discriminant and $\mathcal{O}$ the ring of integers in $K$. Let $x\in K$ (or maybe $\in \mathcal{O}[\frac 1D]$) such that for all primes $p$ in $\mathbb{Q}...

**0**

votes

**0**answers

105 views

### Countable roots of unity

I recently learned about non-standard analysis and have the following question. Take the rational numbers; there is a maximal cyclotomic extension (containing all roots of the multiplicative identity)....

**6**

votes

**2**answers

418 views

### A conjecture involving roots of unity

Motivated by Kevin Liu's recent question, here I pose the following conjecture based on my numerical computation.
Conjecture. Let $m>1$ and $n>1$ be integers. Let $\delta\in\{0,1\}$ and let $\...

**12**

votes

**2**answers

352 views

### $q$ as a prime power and a root of unity

The number of points on the $(n-1)$-dimensional projective space $P^{n-1}(\mathbb{F}_q)$ over a finite field $\mathbb{F}_q$ is the $q$-integer
$$[n]_q := \frac{q^n-1}{q-1}.$$
In analogy, the number of ...

**3**

votes

**1**answer

159 views

### Roots of anti-palindromic polynomial if coefficients are odd.

This is in continuation of the question asked in this earlier post here. Given an anti palindromic polynomial of degree $n$ with odd coefficients, does it have roots on the unit circle?

**3**

votes

**0**answers

156 views

### Modular root of $-1$

Take two distinct coprime integers $a,b$ and take $q=a^2+b^2$. Consider the equation $(xy^{-1})^2\equiv-1\bmod q$. The solutions are two roots which correspond to $(x,y)=(a,b)$ and $(x,y)=(b,a)$. Look ...

**4**

votes

**0**answers

197 views

### power series and roots of unity

Let $p$ be an odd prime and $X$ and $Y$ be subsets of $p^{th}$ roots of unity, $|X|=|Y|=n,X\neq Y.$ Let $f(t)=\sum_{x\in X}x^{t}-\sum_{y\in Y}y^{t}$. If $f(t)=at^k+o(t^k)$ is the power series ...

**5**

votes

**0**answers

106 views

### Sign preserving Galois automorphisms

I have an algebraic number $\alpha \in \mathbb{Q}(\zeta)$, where $\zeta^n = 1$ is a root of unity (not primitive) given as a linear combination of powers of $\zeta$, i.e, $\alpha = \sum_{i=1}^k a_i \...

**6**

votes

**0**answers

167 views

### Infinitude of cyclotomic polynomials with a certain number of terms

Let $\Phi_n$ be the $n$th cyclotomic polynomial:
$${\Phi _{n}(x)=\!\!\prod _{\substack {1\leq k\leq n \\ \gcd(k,n)=1}} \!\!\big(x-e^{2i\pi {k/n}}\big).}$$
Here is a list of the first 30 cyclotomic ...

**4**

votes

**2**answers

353 views

### Summation formulas involving roots of unity to various powers

I want to know properties of the following sum: $$\sum_{j=0}^{p-1} \omega^{\beta j^2}= ~? $$
where $p$ is a prime, and $\omega^p=1$, is a $p$th root of unity (and $\beta$ is an integer between $0$ and ...

**1**

vote

**0**answers

109 views

### How to evaluate this sum of roots of unity with condition to zero

In evaluating the sum:
$$\tag{1}\label{e1}\sum\limits_{j = 1 < h}^N {\left( {{e^{\frac{{i\pi }}{N}\left( {{s_1}j + {s_2}h} \right)}} - {e^{\frac{{i\pi }}{N}\left( {{s_1}h + {s_2}j} \right)}}} \...

**6**

votes

**1**answer

360 views

### Q-binomials at roots of unity

As the title says, given a general $q$-binomial $\binom{n}{k}_q$,
is there some general result regarding its value at a root of unity, $q = \exp(2\pi i r/N)$?

**0**

votes

**0**answers

165 views

### Discrete Fourier transform of the Ramanujan's sums

Let $n$ be a positive integer, and $\zeta_n$ a primitive $n$-root of unity.
I consider the polynomial $P(X) = \sum_{k=0}^{\phi(n)-1} \left[ \sum_{l \in \mathbb{Z}_n^*}^n \zeta_n^{kl} \right]X^k = \...

**2**

votes

**1**answer

143 views

### Simplification of a sum with roots of unity

Let $p$ be an odd prime, $\zeta $ a primitive $p-$th root of unity and
$${a_n}(x) = \sum\limits_{k = 1}^{p - 1} {\prod\limits_{j = 1}^n {\left( {1 + {\zeta ^{jk}}x} \right)} } .$$ It seems that for $...

**4**

votes

**0**answers

124 views

### An Optimization Problem for Exponential Polynomials

Let $\omega$ be a primitive complex $n^{th}$ root of unity. I am interested in the following quantity
$$
\max_{f(n)\leq \ell \leq g(n)} \quad \min_{0<k\leq n-1}
\left| 1+\omega^k+\omega^{2k}+\...

**2**

votes

**1**answer

193 views

### How to prove an approximation of a combinatorics identity

How to prove that $$\sum_{k\ge 0} \binom{n}{rk} =\frac{1}{r}\sum_{j=0}^{r-1}(1+w^j)^n$$ can be approximated as $\frac{2^n}{r}$, where $n\ge 0$, $r\ge 0$, $n>r$ and $w^r=1$ is a primitive $r$-th ...

**4**

votes

**0**answers

215 views

### How small can the nonzero sum of $O(\log n)$ distinct $n-$th roots of unity be?

The OEIS sequence oeis.org/A108380 gives the least number of distinct n-th roots of unity summing to the smallest possible nonzero magnitude.
This sequence seems to imply that the least number of ...

**1**

vote

**1**answer

529 views

### Trace 0 and Norm 1 elements in finite fields

Let $\mathbb{F}_{q^\ell}/\mathbb{F}_{q}$ be the extension of finite filed $\mathbb{F}_{q}$, where $\ell$ be a odd prime and $(\neq q)$. Take $\zeta\in\mathbb{F}_{q^\ell}$. Does there exist different $...

**3**

votes

**1**answer

135 views

### Homomorphism from integral module generated by roots of unity to cyclic group?

Let $S$ be the set of all roots of unity. Consider the $\mathbb{Z}$-module, $\mathbb{Z} S$, as an additive abelian group (that is, $\mathbb{Z} S$ is the subset of complex numbers that can be ...

**7**

votes

**1**answer

917 views

### Uniqueness of sums of roots of unity

Let $\zeta:=e^{\frac{2\pi i}{n}}$, with $n\geq4$, and let $2\leq k\leq n-2$.
Let us suppose that the prime factorization of $n$ is $n=p_1^{\alpha_1}\cdot\dots\cdot p_s^{\alpha_s}$, with $\alpha_i>...

**2**

votes

**0**answers

299 views

### Trace of roots of unity has valuation more than 1 in uramified field

Let $F$ be a finite extension of $\mathbb{Q}_p$ (p is prime) and $K/F$ be a unramified extension of prime degree $\ell (\neq p)$. Denote $\mu_K$ be the group of roots of unity in $K.$ Does there exist ...

**8**

votes

**2**answers

788 views

### Power sums of p-th roots of unity

The following question was asked by a colleague of mine. For any prime $p$ consider
$$ M_p:=\min_{z_1,\dots,z_p}\max_{j,k}\left|z_1^k+\dots+z_j^k\right|,$$
where $z_1,\dots,z_p$ are the complex $p$-th ...

**0**

votes

**1**answer

456 views

### How to calculate $N_{L/k}$(roots of unity)?

Suppose that $L/k$ is a Galois extension of number fields and that $G$
is the corresponding Galois group. Further, for $\frak p$ a prime ideal
of $\cal O$$_L$, let $K=L^{G(\frak p)}$, where [$L$ : $K$...

**3**

votes

**1**answer

293 views

### When is possible to factor a field extension into one which adds no roots of unity, followed by one which adds only roots of unity?

The answer to whether this is possible for general fields is no. However, the counterexamples used two ingredients:
1) $\Bbb Q_p$, whose extensions $K$ containing $\Bbb Q_p(\sqrt[p^e]{u})$ might not ...

**4**

votes

**1**answer

385 views

### Unit in cyclotomic field

Let $n \in \mathbb{N}$ and $\zeta$ be a primitive $n$-th root of unity. I want to know for which $n$ the element $1+2(\zeta+\zeta^{-1})$ is a unit in the ring of integers of $\mathbb{Q}[\zeta]$. Can ...

**6**

votes

**2**answers

436 views

### Why are most coefficients of these minimal polynomials divisible by $p$?

For an odd prime $p$, let $\zeta:=e^{\frac{\pi i}p}$ and choose odd $1<n<p$. Further let $q(x)$ and $r(x)$ be integer polynomials such that $r(x)$ has no common factor with $x^n+1$, and $\xi$ ...

**25**

votes

**0**answers

744 views

### Which sets of roots of unity give a polynomial with nonnegative coefficients?

The question in brief: When does a subset $S$ of the complex $n$th roots of unity have the property that
$$\prod_{\alpha\, \in \,S} (z-\alpha)$$
gives a polynomial in $\mathbb R[z]$ with ...

**4**

votes

**3**answers

219 views

### counting complex roots which are root of unity times a real number

Let $p(x)$ be a monic polynomial over the integers. I want to count the number of roots which have the form $\zeta \cdot r$ where $\zeta$ is a root of unity and $r$ is a real number.
To count the ...

**4**

votes

**2**answers

475 views

### Non-standard Gauss sums

I have the following problem. Let $p$ be some prime. What is the value of
\begin{equation}
\sum_{k=1}^{p-1} \left(\frac{k+1}{p}\right) \omega_p^{kl},
\end{equation}
where $\left(\frac{k+1}{p}\right)$ ...

**15**

votes

**1**answer

1k views

### Evaluating a remarkable term for primes p = 5 (mod. 8)

Let $p > 3$ be a prime number, and $\zeta$ be a primitive $p$-th root of unity. I am interested in knowing the exact value of
$$w_p = \prod_{a \in (\mathbb F_p^{\times})^2}(1 + \zeta^a) + \prod_{b ...

**4**

votes

**1**answer

529 views

### Primes $p=x^2+27y^2$ and Ramanujan's $x_1^{1/3} + x_2^{1/3} + x_3^{1/3}$

I was trying to generalize,
$$\sqrt[3]{\sum_{k=1}^5\cos\big(\tfrac{2^k\cdot\,2\pi}{31}\big)}+\sqrt[3]{\sum_{k=1}^5\cos\big(\tfrac{2^k\cdot\,6\pi}{31}\big)}+\sqrt[3]{\sum_{k=1}^5\cos\big(\tfrac{2^k\...

**3**

votes

**1**answer

174 views

### Does the Lehmer quintic parameterize certain minimal polynomials of the $p$th root of unity for infinitely many $p$?

The solvable Emma Lehmer quintic is given by,
$$F(y) = y^5 + n^2y^4 - (2n^3 + 6n^2 + 10n + 10)y^3 + (n^4 + 5n^3 + 11n^2 + 15n + 5)y^2 + (n^3 + 4n^2 + 10n + 10)y + 1 = 0$$
with discriminant $D = (7 + ...

**9**

votes

**1**answer

327 views

### A curious Gauss-Sum type identity

Let $q=e^{2\pi i/m}$, $a\in\mathbb{R}$ and $1\leq j\leq m-1$. I would like to prove that:
$$(a-1)\sum_{n=0}^{m-1} q^n\frac{\prod_{k=0}^{j-2} (q^{n+k+1}-a)}{\prod_{k=0}^{j} (aq^{n+k}-1)}=0.$$
For $j=...

**4**

votes

**1**answer

403 views

### q-th powers and roots of polynomials

Let $p,q,r$ be integers with $r\ge2$; let $f$ be a polynomial of the form $f(X) = g((X+1)^r)$, which is not a $q$-th power. Let $\omega$ be a $p$-th root of unity.
Prove or disprove that the ...

**0**

votes

**0**answers

87 views

### Condensation points of orbits of roots of unity

For a fixed $n\in \mathbb{N}$ we consider the set of $n$-roots of unity $R(n)=\{z\in S^1; z^n=1\}$. It splits into mutually disjoint orbits. Let $R=\bigcup_{n=0}^{\infty} R(2^n-1)$. For each orbit in $...

**4**

votes

**0**answers

328 views

### Independent units in pure number fields $\mathbb{Q}(\sqrt[p]{t})$

Theorem: In this paper of Frei and Levesque, they correct the proof of a result of Halter-Koch and Stender:
Define the real pure algebraic number field $\mathbb{K}=\mathbb{Q}(\omega_n)$ for $\omega_n=...

**17**

votes

**2**answers

1k views

### Probability a polynomial has a root which is a root of unity

Consider a degree $n$ polynomial $P(x)$ with coefficients $c_i \in \{-1,0,1\}$ chosen uniformly and independently.
What is the probability that $P(x)$ has a root which is a root of
unity?
...

**18**

votes

**4**answers

2k views

### An algebraic number is not a root of unity?

This problem is related to my study of the Burau representation of the braid group $B_3$: I was trying to show that certain "congruence subgroups" are of infinite index.
There is an approach that ...

**5**

votes

**1**answer

707 views

### Minimal polynomial of sums of roots of unity with constant term $\pm1$

Let prime $p$ and given $\zeta_p = e^{2\pi i/p}$. It is well-known that the minimal polynomial of $x = \zeta_p + \zeta_p^{p-1}$ has a constant term either $\pm 1$ and, for certain $p$, the sum of ...

**3**

votes

**1**answer

627 views

### Are all sums of subsets of roots of unity unique?

For a prime $p$, let $S$ be the set of all $p$-roots of unity in the complex plane. Now, consider the sum, $W(R)$ of the members of a set $R$ which is a proper subset of $S$. I suspect that $R \ne R'$ ...

**24**

votes

**3**answers

3k views

### Products of primitive roots of the unity

Let $m>2$ be an integer and $k=\varphi(m)$ be the number of $m$-th primitive roots of the unity. Let $\Phi = \{ \xi_1, \ldots, \xi_{k/2}\} $ be a set of $k/2$ pairwise distinct primitive $m$-th ...

**3**

votes

**1**answer

280 views

### Diophantine equation with primitive nth root of unity

Fix an $n$th primitive root of unity $\xi$. I need to understand if we can characterize in an easy way all the solutions $k \in \mathbb{Z}$ of the equation $\left|1-\left(-\frac{\xi^k - 1}{\xi-1}\...

**5**

votes

**3**answers

3k views

### Can the sum of two roots of unity be a root of unity?

Let $p$ be prime, and $z_0, z_1, ..., z_{p-1}$ be all the $p$-th roots of unity, i.e. solutions of the equation $z^p = 1.$
Is it true or false that a combination of two (or more, in general) of the ...

**1**

vote

**1**answer

3k views

### Primitive $k$th root of unity in a finite field $\mathbb{F}_p$

I am given a prime $p$ and another number $k$ ($k$ is likely a power of $2$). I want an efficient algorithm to find the $k$th root of unity in the field $\mathbb{F}_p$. Can someone tell me how to do ...

**2**

votes

**0**answers

110 views

### Rational viewing points in a polygon

We refer to the question posed in Seeing the vertices of a polygon with rational angles, but now ask for constructions or for the existence of rational viewing points. We'll call a point $p$ inside (...

**6**

votes

**1**answer

1k views

### Sum of products of p-th powers of roots of 1 and monomial symmetric functions

Hello mathematicians,
i'm looking for explicit computations of expressions like
$$
\sum_{\substack{0\leq i,j,k<n\\i\neq j\neq k \neq i}}\zeta_n^{ip^{k_1}+jp^{k_2}+kp^{k_3}}
$$
and its ...

**0**

votes

**1**answer

1k views

### nontrivial cube root of unity [closed]

Hi,
I have a finite field Fp with p = 11 mod(12) and I am trying to get the third nontrivial root of unity in Fp^2 = Fp^2[x]/(x^2+1). So, i need x where x^3=1.
Somehow I came into a source saying ...