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# Questions tagged [roots-of-unity]

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### 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

• 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

• 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

• 171
6 votes
2 answers
511 views

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 ...
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

• 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

• 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

• 253