Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [roots-of-unity]

The tag has no usage guidance.

11
votes
2answers
160 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
1answer
113 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
0answers
145 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
0answers
148 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
0answers
102 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
0answers
154 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 ...
2
votes
0answers
154 views

Vanishing sums of roots of unity

Lets say I have a multivariate polynomial $P(x_1, \ldots, x_n)$ of degree at most $d = d(n) = n^c$ for a fixed constant $c$. I know that the Kronecker map ($x_i \to x^{d^i}$) preserves zeroness/...
4
votes
2answers
229 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
0answers
94 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)}}} \...
5
votes
0answers
203 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
0answers
141 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 = \...
1
vote
1answer
119 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
0answers
119 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
1answer
189 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
0answers
188 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
1answer
389 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
1answer
132 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 ...
6
votes
1answer
753 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
0answers
278 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
2answers
599 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
1answer
295 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
1answer
279 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
1answer
325 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
2answers
427 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 $\xi$ ...
23
votes
0answers
657 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
3answers
197 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
2answers
444 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
1answer
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
1answer
513 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
1answer
166 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
1answer
312 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
1answer
392 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
0answers
86 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
0answers
325 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
2answers
983 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
4answers
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
1answer
597 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 ...
2
votes
1answer
528 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
3answers
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
1answer
263 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
3answers
2k 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
1answer
2k 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
0answers
107 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
1answer
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
1answer
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 ...
80
votes
4answers
11k views

How small can a sum of a few roots of unity be?

Let $n$ be a large natural number, and let $z_1, \ldots, z_{10}$ be (say) ten $n^{th}$ roots of unity: $z_1^n = \ldots = z_{10}^n = 1$. Suppose that the sum $S = z_1+\ldots+z_{10}$ is non-zero. How ...
0
votes
0answers
477 views

Looking for product of symmetric polynomials evaluated at roots of unity

Consider $a_{1} = \alpha^{N-n}, a_{2} = \alpha^{N-n+1}, a_{3} = \alpha^{N-n+2}, a_{4} = \alpha^{N-n+3}, \cdots, a_{n} = \alpha^{N-1}$ where $\alpha$ is a complex $N$th root of unity where $N = 2 + (n-...