Questions tagged [roots-of-unity]
The roots-of-unity tag has no usage guidance.
71
questions
1
vote
0
answers
113
views
Monomial symmetric polynomials evaluation at roots of unity
The monomial symmetric polynomials are defined see Wikipedia.
For an arbitrary partition $\lambda$ with $n$ parts
I'm trying to find the following values:
$$m_{\lambda}(\omega_0,\dotsc,\omega_{n-1})$$
...
- 53
4
votes
1
answer
221
views
Third roots of unity and norm element
Let $K = \mathbb{Q}(\zeta_3)$ where $\zeta_3$ is a third root of unity, let $F = \mathbb{Q}(\sqrt[3]{\ell})$ where $\ell \equiv 1 \pmod{9}$ is a prime and set $L$ to be the Galois closure of $F$, i.e.,...
- 1,251
1
vote
0
answers
35
views
Finding the radical expressions of trig functions [closed]
I am trying to find the exact radicals of the sine and cosine of (m/n)*pi. I have been using sympy to do this and made this pretty good script:
...
4
votes
0
answers
146
views
Vanishing exponential sums of fractional parts of polynomials
Let $p$ be an integer polynomial and $k$ be a natural number, both fixed. Is it the case that if
$$C(\alpha) = \sum_{i=1}^k e(\alpha \{p(i)/k\})$$
equals 0, then $\alpha$ is an integer? Here, $e(x) = ...
- 41
2
votes
0
answers
235
views
Sum of roots of unity
Today I came across the series $\sum_{k=0}^{n-1}\varepsilon^{2k^2}$, where $\varepsilon$ is some primitive $n^\text{th}$ root of unity. Is there an explicit expression for this sum? I mistakenly ...
- 10.4k
1
vote
1
answer
174
views
Norm of $2^{i}$-th primitive root
Let $ K $ be finite degree extension of $ \mathbb{Q} $ such that $ -1 $ is not a square in $ K $. Let $ L = \frac{K[x]}{\langle x^2 +1\rangle}$. Thus every element of $ L $ is of the form $ a + ib $ ...
- 913
1
vote
0
answers
53
views
About nilpotent Jordan algebras, matrix representations and formally real algebras
Given an non-commutative associative unital algebra A of characteristic $0$, one can construct a Jordan algebra $A+$ using the same underlying addition vector space.
Notice first that an associative ...
- 763
1
vote
0
answers
123
views
$\sin(\frac{\pi}{p}) $ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$?
We have the following identities:
$\sin(\frac{\pi}{1})=0$
$\sin(\frac{\pi}{2})=1$
$\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$
$\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$
Lets start with a definition.
Rules ...
- 763
3
votes
1
answer
331
views
A similar relationship between the generic cubic and the Lehmer quintic?
I. Comparison
It doesn't seem to be well-known that the generic cubic (prominent in this MO post) for $C_3 = A_3$,
$$x^3-nx^2+(n-3)x+1 = 0$$
has the nice property that its roots $a,b,c$, if in correct ...
- 12.2k
5
votes
1
answer
311
views
A conjectural permanent identity
Let $n>1$ be an integer, and let $\zeta$ be a primitive $n$th root of unity. By $(3.4)$ of arXiv:2206.02589, $1$ and those $n+1-2s\ (s=1,\ldots,n-1)$ are all the eigenvalues of the matrix $M=[m_{jk}...
- 15.1k
2
votes
1
answer
502
views
Integer eigenvalues of a class of matrices inspired by Prof. Zhi-Wei Sun's conjecture
Theorem: Let $n>1$ be an odd number and $\zeta$ a primitive $n$-th root of unity. Then
\begin{eqnarray}
&&\sum_{\tau\in D(n-1)}\mathrm{sign}(\tau)\prod_{j=1}^{n-1}\frac{1}{1-\zeta^{j-\tau(j)...
4
votes
1
answer
580
views
Has any one seen this sum of roots of unity before?
Fix a prime $p >2$ and $q_1$, $q_2$ such that $q_i - 1$ is exactly divisible by $p$. For any $n$, $a$, $b $, consider the sum
$$\sum_{i=0}^{p^{n-1}-1}\zeta_{p^n}^{aq_1^i+bq_2^i}.$$
Is this always ...
- 7,716
7
votes
0
answers
148
views
Finding when a certain product in a cyclotomic field is equal to one
For the following, fix $m\geq 2$, and let $a_{0},\dots,a_{m-1}\in\mathbb{Z}$ be such that $\sum_{k=0}^{m-1}a_{k}=0$. I would like to find the exact conditions on the $a_{k}$ so that the following ...
- 299
15
votes
2
answers
1k
views
Vanishing of a sum of roots of unity
In my answer to this question, there appears the following sub-question about a sum of roots of unity. Denoting $z=\exp\frac{i\pi}N$ (so that $z^N=-1$), can the quantity
$$\sum_{k=0}^{N-1}z^{2k^2+k}$$
...
- 52.1k
3
votes
0
answers
193
views
Coefficients for Expansions of $1-\zeta_p$
Let $\mathbb{Q}_p(\zeta_p)$ be the cyclotomic extension of the $p$-adic field $\mathbb{Q}_p$. Then $1 - \zeta_p$ is a uniformizer for this field. Recall that
$$\sum_{i=1}^{p-1} \zeta_p^i = -1.$$
So ...
- 31
1
vote
1
answer
218
views
A determinant involving the cotangent function
Let $n>1$ be odd. In my 2019 preprint On some determinants involving the tangent function, I proved that
$$\det\left[\tan\pi\frac{aj+bk}n\right]_{1\le j,k\le n-1}=\left(\frac{-ab}n\right)n^{n-2}$$
...
- 15.1k
10
votes
1
answer
556
views
Identities involving derangements and roots of unity
For a positive integer $n$, a derangement of $\{1,\ldots,n\}$ is a permutation $\tau$ of $\{1,\ldots,n\}$ with $\tau(j)\not=j$ for all $j=1,\ldots,n$. For convenience, we let $D(n)$ denote the set of ...
- 15.1k
11
votes
1
answer
634
views
A conjecture on binomial coefficients and roots of unity
Is the following true?
Let $p$ be a prime and let $w$ be a $(p-1)$st root of unity (not necessarily primitive). Then
$$\binom{w}{n}=\frac{w(w-1)\cdots(w-n+1)}{n!}$$ is $p$-integral; i.e., it can be ...
- 16.8k
1
vote
0
answers
90
views
Mod $N^2$ evaluation of a polynomial defined by first $N-1$ roots
Given a prime $N$ and integer $g$, where $g$ is able to generate the multiplicative subgroup $(\mathbb{Z}/N^2\mathbb{Z})^*$, I am interested in any results simplifying or evaluating $f\in (\mathbb{Z}/...
- 11
6
votes
0
answers
124
views
Simultaneous vanishing $\mathbb{Q}$-linear relations between $N$-th roots of unity
Let $\zeta$ be a primitive $N$-th root of unity and $\Gamma \subset (\mathbb{Z}/N)^\times$ a subgroup. Let $|\Gamma|$ be the cardinality of $\Gamma$ and consider the linear map $M_\Gamma\colon \mathbb{...
- 171
2
votes
1
answer
339
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 ...
- 171
8
votes
0
answers
352
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 <...
- 191
2
votes
0
answers
233
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 ...
- 21
5
votes
0
answers
272
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}...
- 171
6
votes
2
answers
511
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 $\...
- 15.1k
13
votes
2
answers
629
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 ...
- 1,430
3
votes
1
answer
330
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?
- 657
3
votes
0
answers
160
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 ...
- 13.8k
4
votes
0
answers
299
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 ...
- 205
5
votes
0
answers
119
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 \...
- 305
6
votes
0
answers
229
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 ...
- 615
4
votes
2
answers
745
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 ...
- 253
1
vote
0
answers
130
views
How to evaluate this sum of roots of unity with condition to zero
In evaluating the sum:
$$\tag{1}\label{e1}\sum\limits_{j = 1 < h}^N {\left( {{e^{\frac{{i\pi }}{N}\left( {{s_1}j + {s_2}h} \right)}} - {e^{\frac{{i\pi }}{N}\left( {{s_1}h + {s_2}j} \right)}}} \...
- 59
6
votes
1
answer
673
views
Q-binomials at roots of unity
As the title says, given a general $q$-binomial $\binom{n}{k}_q$,
is there some general result regarding its value at a root of unity, $q = \exp(2\pi i r/N)$?
- 15.8k
0
votes
0
answers
209
views
Discrete Fourier transform of the Ramanujan's sums
Let $n$ be a positive integer, and $\zeta_n$ a primitive $n$-root of unity.
I consider the polynomial $P(X) = \sum_{k=0}^{\phi(n)-1} \left[ \sum_{l \in \mathbb{Z}_n^*}^n \zeta_n^{kl} \right]X^k = \...
- 313
2
votes
1
answer
188
views
Simplification of a sum with roots of unity
Let $p$ be an odd prime, $\zeta $ a primitive $p-$th root of unity and
$${a_n}(x) = \sum\limits_{k = 1}^{p - 1} {\prod\limits_{j = 1}^n {\left( {1 + {\zeta ^{jk}}x} \right)} } .$$ It seems that for $...
- 5,592
4
votes
0
answers
150
views
An Optimization Problem for Exponential Polynomials
Let $\omega$ be a primitive complex $n^{th}$ root of unity. I am interested in the following quantity
$$
\max_{f(n)\leq \ell \leq g(n)} \quad \min_{0<k\leq n-1}
\left| 1+\omega^k+\omega^{2k}+\...
- 10.3k
2
votes
1
answer
211
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 ...
- 167
4
votes
0
answers
274
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 ...
- 10.3k
1
vote
1
answer
788
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 $...
- 255
3
votes
1
answer
197
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 ...
- 61
7
votes
1
answer
2k
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>...
- 73
2
votes
0
answers
377
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 ...
- 255
8
votes
2
answers
1k
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 ...
- 103k
0
votes
1
answer
959
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$$_L$, let $K=L^{G(\frak p)}$, where [$L$ : $K$...
- 197
3
votes
1
answer
361
views
When is possible to factor a field extension into one which adds no roots of unity, followed by one which adds only roots of unity?
The answer to whether this is possible for general fields is no. However, the counterexamples used two ingredients:
1) $\Bbb Q_p$, whose extensions $K$ containing $\Bbb Q_p(\sqrt[p^e]{u})$ might not ...
- 197
4
votes
1
answer
523
views
Unit in cyclotomic field
Let $n \in \mathbb{N}$ and $\zeta$ be a primitive $n$-th root of unity. I want to know for which $n$ the element $1+2(\zeta+\zeta^{-1})$ is a unit in the ring of integers of $\mathbb{Q}[\zeta]$. Can ...
- 41
6
votes
2
answers
472
views
Why are most coefficients of these minimal polynomials divisible by $p$?
For an odd prime $p$, let $\zeta:=e^{\frac{\pi i}p}$ and choose odd $1<n<p$. Further let $q(x)$ and $r(x)$ be integer polynomials such that $r(x)$ has no common factor with $x^n+1$, and $\xi$ ...
- 13.3k
26
votes
0
answers
907
views
Which sets of roots of unity give a polynomial with nonnegative coefficients?
The question in brief: When does a subset $S$ of the complex $n$th roots of unity have the property that
$$\prod_{\alpha\, \in \,S} (z-\alpha)$$
gives a polynomial in $\mathbb R[z]$ with ...
- 1,513
4
votes
3
answers
303
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 ...
- 253